Hi there! This is actually a relatively low-budget set up. :) I got this used Yamaha DTX400K from Guitar Center 3 years ago for $200. I rearranged the toms as a 1-up 2-down set up just as a preference. I'm using the stock sounds included with the kit but I made a custom configuration with Yamaha's app using my phone connected to the kit via USB. I plugged the headphone output directly into a old Tascam DR-1 portable recorder with Instant Destiny loaded on to it, then I just used the "overdub" feature to play over the song. Back in the day I think the DR-1 was like $200 bucks but newer, cheaper portable recorders can do the same thing. This one looks cool! Has 2 track recording too, which my DR-1 doesn't support.
TL;DR: If you're patient and willing to buy some used gear, I think it's relatively easy to keep your expenses below $500 with a minimal set up. :)
https://www.amazon.ca/Yamaha-DTX-400K-Electronic-Drum-Kit/dp/B008RWHKJ2 this is around 500 and you don't have to worry about noise.