I reckon if you’ve got the 9sp LX, just use that. Especially if your front is going to go 48 to 22.
Having said that, 44/32/22 is probably fine. If you’re pumping away at 44/11 trying to go faster just stop pedalling and enjoy yourself. There’s a small chance your Force FD will shift 9sp over 2 cogs too, so maybe you could lose the outer ring and stick say a 42t on the middle for a 42/22. I’d go 40/22 no worries but I think the idea of that might freak you out.
No, forget this idea. If you want a bigger ring, get something like a 50/34.
What model deore?
RD-M592-SGS (long, shadow) has 45T capacity and 36T max sprocket. Ok for 44T big ring, but not 48T.
RD-T661 (long, non shadow) has 43T capacity and 32T max sprocket. Ok for 44T big ring, but not 48T.
Alivio RD-M4000 shadow is cheap and has a barrel adjuster built in, 45T capacity, 36T max sprocket.
For the FD you’ll need something like the FD-3503 / FD-5703 (50T max, 20T difference) if you want the indexing in the sora shakes to work, but if you used a downtube shifter (mikeD current jam) for the front you could use any old MTB triple mech and it’ll have a better cage profile for the 44T big ring.
Another option is running an older set of 10speed Campag shifters as 9spd shimergo. Currently running a pair of centuar shifters on the GF’s bike with a 9spd lx fd and 3x cranks. The many points of indexing on campy makes setting up the fd real easy