Just stumbled upon this searching for something else so I expect you've already travelled. In case you haven't, I regularly do a similar trip to what you describe.

When I check in, I get boarding passes for the EK and FZ flights.

When I arrive at T3, I go through transfers to T3 and proceed to the Ahlan lounge at B. A few hours before my FZ flight, I follow the signs to T2. It does involve going backwards through one of the transit security areas but it's well signposted and the staff are used to it. When I go through at around 9-10am, there's nobody coming the other (normal) way so it's very easy. This leads to an area to wait for a bus for T2.

Incidentally, I find the toilets when you arrive at T2 by bus before going through transit security are the best in DXB and try to plan to do any major ablutions there. After security, the toilets nearly always have a queue out the door. The toilet in the Ahlan lounge is terrible.

The Priority Pass/Marhaba lounge in T2 is awful and always very crowded when I've visited in the past.

Also, if anyone can help, I'm trying to figure out if it's possible to go from T2 to T3 with a ticket for a flight departing from T1. I've found myself needing to grab some clothes for a trip but I can't recall any clothes shops at T1 so was hoping to pick up a few things at T3 if possible.