The control board inside a 3 is the same control board inside a 5 or a 6, so it will certanly connect. You are going to not want to update your firmware though, because the 4, 5, and 6 firmware uses a different leadscrew z steps value. As long as you connect and tell it to just use the existing firmware, you should be fine. you’ll have to adjust your print profiles to get the temperatures right for the older style nozzle, but it should otherwise work.
I would reccommend you look into sourcing a set of the Taz 4-6 style leadscrews and upgrade your printer to at least a Taz 5 spec. The new hotend and Z axis setup really improves quality oveer what the 3 is capable of. The endplates will run you $40 https://itworks3d.com/product/lulzbot-taz-4-5-x-axis-endplate/ Finding the leadscrews and leadscrew nuts is a bit tricky, but the ones for the 6 work just fine there too. You also need the two couplers, 4 608zz bearings, and the parts for the Z endstop height adjustor(m5 heat set insert, bolt, spring, thumbscrew end), the X axis limit switch extension wire, or the auto leveling parts for the 6 (expect to spend $150 there)