translation; 9s MTB derailleur, An LX model...?
In which case yes I agree that will work, any 9s shimano MTB rear mech will shift OK using your present STIs.
In theory road 9s STIs pull fractionally different amounts of cable than MTB 9s ones in the middle of the cassette, which is (I think) why shimano don't endorse this combination in their compatibility tables (the break in the HG ramping is different in 'road' and 'MTB' 9s cassettes, and this shifters are different to suit the cassette, rather than the RD). However in practice plenty of people run this setup and they don't have a problem. Until very recently you could still buy an XT 9s mech, new, but they seem to have run out now, so there is a more limited choice of decent 9s MTB mechs.
Anyway if you fit one this should get you to a 36T sprocket....?
Also most compact doubles with 110mm BCD five bolt pattern will accept a 33T inside ring (eg made by TA) although in some cases you may have to file the ends of the spider arms slightly to get enough clearance for the chain.
34/32 is 28.7" and at the same wheel size 33/36 is 24.7", which is a worthwhile reduction in the bottom gear ratio.
cheers