Using the FT-2900 for APRS is fine, you can tuck it away out of sight once configured and forget about it. There's no need to watch the radio on APRS, although remember to lock the controls so the frequency can't be changed.
Yes, the FTM-350 can run APRS in the background with regular use on the other side. When it needs to transmit an APRS packet it blanks the other side. If you are currently transmitting it will skip the packet and if you are receiving it goes quiet for a second or so. This is an improvement over the FT-8800/8900, which can listen to two bands at the same time but can only transmit on the selected side. So if the APRS tracker wants to TX, it will do it on the frequency you're using currently and not the secondary side.
It sorta depends on which radios you use, but if you are starting from zero going with a FTM-350 or TM-D710A might be competitive, particularly if you are looking at a FT-8800 or other fairly expensive radio already. If you have any old junk, probably cheaper to build your own. You can use one of those Chinese HTs for APRS, potentially a whole APRS set up might be $100 ~ $150. You'd also need to run two antennas, which can also be a pain.
BTW, the FTM-350 is discontinued, so existing stock is all that's available. Yaesu came out with the FTM-400, which is a 350 with digital modes added. Not sure the cost, but probably not cheap.
"Silent enim leges inter arma." -- Cicero