Build Concept :: Light weight and simplicity mean performance, regardless of budget.

The AUW for this quad is <400g with lighter battery choices, and 425g specced for a 2 minute race.
For tighter tracks, this outperforms my larger and faster quads, just because light weight for the same disc loading ultimately is better than a higher thrust to weight ratio of more powerful setups.

For anybody who wants to get into lightweight 5" racing quads but doesn't want to have to commit a lot of money to it, this is a useful starting reference on what can be done. If budget is that tight I still actually recommend a very similar 3" racing build (with a lot of these same parts) just because they can run cheaper batteries and will be a bit more durable if you're just starting off flying through gates at speed... but you can use the same batteries.

Frame: The Hyperlite Tooth Fairy is a big part of how this achieves the budget. It's a solid frame to build a lightweight racer out of, and performs quite well. The best version for performance and durability is carbon top & bottom plate, but I can't help but get a matching purple canopy, this one is from DroneBuildz, which happens to house the VTX and antenna quite nicely.

Motors: RCX Motors have been the ultimate answer for best performance on a tight budget. These H2205's are going to be the top of thrust tests or bling meters, but the performance you get is impressive. Older construction (flat magnets, smaller shafts, smaller bearings) average out to being middle of the road for smoothness and not having quite the top end power, but with the latest propellers and by running the 2633KV version, the power is still there. To hit the overall weight target, 2205 is the right size - any smaller and you end up restricted on what props really work well, any larger and the weight adds up and durability of these 4mm arms starts to suffer. Similar motors will also get it done, the hard part is finding sales... but 2205/2205.5/2206 motors are always impressive despite their drawbacks. The RCX SE2205's offer even more power for $9/motor, although that extra power isn't necessarily something you'd want to run through a cheap stack. The H2205's in 2350KV are also great, but you'll want to run the DAL T5051C or Ethix S4 props to make those come alive.
While I'm listing off the limitations, keep in mind that these were $7 motors. Seven. Dollars.

FC & ESC: I opted to run the Diatone Mamba F25 stack. The F4 FC is actually great for the money, comes with a very usable 16MB BBL flash, and plugs right into the F25 ESC. This ESC shares a lot in common with the Airbot Ori 25A (BLHeli_S version), and is really better suited to use in a 3" quad that runs 4S... but with the right prop selection and making good use of newer Betaflight features you can stay within its performance envelope really well. For a very budget stack, the overall ease of build and very logical pin layout works very well. There are a lot of design features shared with the pricier Diatone Mamba stacks, but this one is clearly built to a price point, so I wouldn't recommend running much more through these electronics than this.

FPV - Camera and VTX: For this, budget and simplicity carries the day. The Foxeer Razer Micro camera is great, and definitely the right price for this particular budget target. The AKK Race VTX is also excellent, and pairs quite nicely with the EMAX Stubby Nano antenna for an overall extremely light setup. Even ignoring that these three parts are some of the most cost-conscious parts of the build, I wouldn't hesitate to run this trio in my race quads where budget wasn't a consideration - for racing these just work, so this will be what I bring with me to races. The only marginal difference in performance I can notice is the camera defaults for the Predator Micro are better, but the difference is pretty slight. For the VTX and Antenna, there is no difference I can identify.

Propellers: On a budget, it's hard to ignore either bulk pricing (20 sets) from Pyrodrone, or the HobbyCool pricing on DAL T5145C props. They all happen to come in my favorite color for racing quads (51433 and T5145C in purple), so that's just extra icing. The performance feel and relatively light loading from these is excellent. Other options would include the Ethix S3, DAL Nepal, HQ 5.1x3.6x3, Azure 5140, and TMotor 5143 for being light loading but very lightweight props.
Honorable mention goes to the DAL 5050c Biblade props - overall performance is amazing, but not really beginner-friendly (lots of top end speed and the cornering performance isn't compromised because of how light this setup is - it carries speed amazingly, but it's harder to fly because it always wants to carry speed).

Radio Control: The cheapest receiver I like to run is the XM+, and it works great. Hobbycool is a consistent source to find these at a really good price, so I tend to get these and propellers from there. I'm partial to running a pair of zip ties off the arm and heat shrink attachment of the antenna active elements to that, for solid range and keeping them out of props in crashes. This hasn't necessarily been that successful, and cold weather crashes do tend to result in broken zip ties, but this is my preferred setup.

Batteries: Since this is a 4S stack, 4S batteries are what work best. The cheapest I can find consistently are the CNHL 4S 1300mAh batteries, and on sale ($12-14 direct from CNHL) that's an unbeatable deal. The 1000mAh packs also work extremely well, and the bonus is that they're lighter. It's entirely possible to race with 850mAh packs if you're either disciplined on the throttle, or trying to get maximum pace and agility for a shorter period of time. Importantly, these are all very affordable battery packs. With lightweight builds that aren't centered around peak power, the peak amp draw of the battery isn't as critical, and this keeps 4S builds viable for racing pace.

Firmware & Tuning: Betaflight 4.1 with RPM Filtering (available with JESC 2.3 for 48kHz awesomeness) results in a ridiculously smooth flying setup that is absolutely locked in, with at most a couple of slider adjustments. Since you can quite easily grab blackbox logs, you can get help from the UAVTech Discord or Betaflight BBL Facebook to get it truly dialed in, but this flies so well out of the box it's not required at all - just make sure that the RPM telemetry is working correctly after you reflash everything, and go fly.
Especially with setups that are a bit on the edge, throttle limiting is very much a useful part of this - I run mine at 90% throttle max from the transmitter, and this is enough that running 57° camera angle is where this quad feels most natural (for me). I realize this is a lot of camera angle, but the one thing about very light quads is that they actually will decelerate somewhat in a straight line (aero drag acting on a reduced mass) compared to heavier quads, so the flight style is a bit more point & squirt than a more drift & flow approach.

Budget: My particular iteration is a $148 quad, with visual accessories. As originally built, I did actually hit the $132 price. This is very much a budget targeted build, but the list of compromises is extremely short. With smaller and lighter motors, you can get solid performance from a budget ESC, and a very minimalist frame/electronics stack means the performance is all there.
I spent a touch more to add a canopy (run the stock frame for better everything but aesthetics), and the LED panel on the back, but this is still a <$150 quad that lets me keep up with pilots flying gear at twice this price. It's possible to trim $8 by going to a VTX03S VTX (for the same price) and using its included dipole antenna, but I do feel that having an RHCP/LHCP capable quad is worth it if you're going to organized races.. Otherwise, if you want to spend less, you will either give up reliability, capability, or need to go to 3". The really good news is that both this, and my even more budget-oriented 3" quads can share batteries (4S 850mAh packs) that are quite cheap when on sale, which helps, although it isn't optimal race performance on either setup.