I haven't seen anything that fits those requirements in that price range.
Just getting a PixHawk flight controller so you can program the drone would cost that much. Same for a good radio transmitter to actually fly the drone.
I think DIY is out of the question at that price point.
That's a nice controller, but heavy and expensive, 37 grams and $75 according to amazon. My sub-250g build used a Naze32 which weighs about 4 grams without connectors and was $35. I flashed it with cleanflight from github
https://github.com/cleanflight/cleanflight
Presumably OP wanting to modify stuff will want FOSS like cleanflight.
This is of course ancient obsolete history because things move fast in multicopter world and that was like 6 months ago. Not even sarcasm. This advice was reasonably contemporary last fall, but this spring I donno.
OP will rapidly get into a scalability problem where I assume OP wants to do something interesting or complicated and aerospace is highly optimized so OP is likely to have much more success driving a flight controller as if it servos using a ras-pi or something, than trying to compile cleanflight to add Alexa to the cleanflight itself or whatever it is OP is trying to do.
Electronics have improved quite a bit. Back in the 80s we'd buy a $200 mosfet motor controller with a half pound heatsink and connect it to a $20 motor in your RC dune buggy (running off old fashioned nicads LOL) but now a days you attach a $11 no-heatsink controller to a $40 brushless motor and call it good. Of course you don't need a cooling fan if you have hundreds of watts of fan blowing on it from half an inch away...
Something to think about ops desire for a $100 multicopter is control electronics and surprisingly chassis and props and serial receivers are cheap, but I dropped $40 on EACH motor, by far the most expensive part of the build. Of course at 220 grams flying weight my build has the vertical acceleration performance of a large model rocket, which can be slightly terrifying when learning to fly, something lower performance but still flyable might be cheaper. I just thought it interesting that most of my build's money is in the motors, like about half of it.
Pilot induced oscillation, never fun.
I can hover around 1/3 throttle so that would imply 1 G of thrust so you can guess that flooring it would be 2 G vertical acceleration. So 64 ft/s squared so one second after I floor it I'm going 40 mph straight up.. It feels like throwing a $300 volleyball as hard as you can straight up.
To stay underweight I use small 500 mAH lithium poly batteries which lead to both high acceleration and short flight times, like 3-4 minutes tops. I think I have a 250 somewhere which is really small and light. I could probably physically lift 2000 mAH batteries or larger, but would then be over the FAA 250 gram limit so I'd have to register as a drone pilot, which I'm OK with, other than I haven't done it yet. Great, another government list to find myself on, just what I needed. Once you're on enough lists, one more won't hurt...