This is incorrect.

When you arrive to T3 Flight Connections, you pass through security, then pass the airline desks, and end up near the lounges. If you do this, look above you as you walk into the departures area and you'll see that the passageway you've just come through is signed with a big purple Flight Connections sign. It's absolutely possible and permitted to enter T3 via Flight Connections with no T3 boarding pass, do whatever you want to do in the T3 departures hall, then follow the purple Flight Connections sign from the T3 departures hall and head back to the Flight Connections area to board a bus for T5. At this point you'll need a valid boarding pass for T5, as unlike T3 it is necessary to have a valid T5 boarding pass to enter the T5 Flight Connections security area and onward to the T5 departures hall, but that's not what the OP said was desired.

The question was whether it's possible to arrive to T2, use a lounge in T3 then leave on a flight from T5, and from the perspective of security and passenger flow this is indeed possible and is signed.

This is a valid question; the OP indicated that the desire was to use a paid lounge at T3... I'd imagine a paid lounge would sell admittance irrespective of one's onward flight, however I suppose there's no stopping them from asking for a BP and turning you away. I've not done this so would have to defer to someone who's bought an Admirals Club day pass at LHR recently and can confirm whether they do any such checks.