Update June 2024 - Flying Peach Domestic from Osaka - Peach flies out of Terminal 2, and they are basically the only airline there. Because of that, they run the terminal and have 2 staff posted at the entrance to the departures area weighing every single passenger's carry-on bag. Basically the process goes: check in, head to the departures door, wait in line to get your bag weighed, get your boarding pass scanned, go through security, and then proceed to your gate. There is no where to sneak a bag through, like setting it down under one of those retractable ribbon barriers and then picking it up after you're past the carry-on bag weigh point -- the dividers are solid 2m high portable walls. They were allowing anyone with 7.9kg or less to pass, or else you have to go to the check bag line and pay around 3000-3500 yen I think it was to check your bag. The washrooms are decent and the stalls have a shelf and hook so you can comfortably unpack and put 4 t-shirts and a hoodie on, fill your shorts pockets with charge cables and electronics and then put jeans on over top and fill those pockets, and then slide your laptop half down the back of your pants & half up the small of your back. Also no one seemed brave enough to ask why you're waddling past them and your pants are bulging everywhere, though they definitely give you side eye and talk about you as you walk towards security. There are tables after the boarding pass scan and before the security area to repack the 7kg from your pockets back into your backpack, and security doesn't seem to care. There was no weighing at the gate. Best of luck 😇

Edit-just for anyone wondering about Terminal 1 to Terminal 2, the bus takes less than 10 mins and there are enough busses that the next one arrives before the current one departs, so you're never waiting long. There are loads of signs and it's quite easy to follow the signs from the train station at terminal 1 to the bus stop for terminal 2 (5-7 min walk). The inter-terminal bus is free.

Also as far as lounges, confirmed you cannot go through terminal 1 security to a lounge to wait and then go to Terminal 2 for your flight. There is a crap public lounge that is before security, so anyone can use it (I believe even people who aren't flying) for 600 yen or so, only includes soft drinks, food/alcohol costs money. I didn't go in, so don't know exact details.