You’re talking about having an active flight controller on the board to maintain attitude, not just move servos at specific times, right?
I flew my Quicksilver (RIP) with a Motolab Cyclone quadcopter flight control board. It’s cheap & certainly capable of doing that. It has gyros, accelerometers (up to 16Gs, so watch that), barometer, a micro SD card slot for data logging and a very capable CPU (STM32 F3). It’d take you doing a little bit of coding for the rocket-specific flight controls, but if you have those skills it’d be a straight forward project. With the typical gyro/accel16G limit on them, it’d be worth flying them without active controls first to make sure the orientation data and your control inputs look good.
There are tons of options for quadcopter flight control boards & most are under $50 with many in the $20 range. Look around for one with the capabilities you need. With some fairly minor mods, you could even control ejection charges.
I used mine with iNav software, a GPS, and a telemetry radio (FrSky XSR). I had real time telemetry on my ground station and pretty cool flight data.