Joshua,
Have you had any trouble with RF energy playing havoc with a nearby microprocessor that is not shielded? I ask because I've got an unassembled amplifier kit and before I put it together, wired for a single band, your experiment has me thinking. Using either an official Arduino UNO or one of Adafruit's clones with a prototyping shield installed, I could wire up the transistors (on the shield), drawing power for them and the relays from the board. Placing the Arduino + shield combination and the LPF unit in a project box, my wiring connections would be reduced to RF and PTT from QDX to amplifier, two coax connections to/from the project box and the amplifier, a 3-pin (1/8" TRS) connection for band selection (QDX to project box), and then a DC to power the Arduino, etc (and the amp and the QDX!). My key concerns are, first, grounding - I've no desire to create any destructive ground loops! - and, secondly, since the (amplified) RF would be going through an LPF that is closely located to the unshielded (no pun intended) microprocessor...well, I don't want to transmit on 20m and have the microprocessor select the 40m LPF ;-)
This experiment of yours is of great use to me, as I operate FT8 on 40/20/15m (I don't band-hop?per se, but...) and your solution would make it so much easier to do so with amplification!
Regards,
Charles Johnson
KF4AYT