I'm no expert on Flat track, but I'd guess tires at a minimum?
Probably wheels to conform with rules and be more competitive.
In Australia on our looser "Dirt Track" circuits people tend to run 21" front knobbys and a trials tire in 18 or 19".
On the firmer tracks 19" front and rear with dirt track tires. You seem to be able to buy 17" Dirt Track tires if that is an option.
I think the set up would be similar so one bike for both styles would seem ok.
(A lot less compromise/work than try to use a bike for Sumo and MX.)
Wheels/brakes and maybe bars (are aluminium "bark busters" allowed in Flat Track?) and you'd have 2 great bikes!