As an experienced PlayStation® 3™ speedrunner of this game, FDTs that require a lot of fps (90+) are not viable for these consoles, however, FDTs that require game lag - like A Poisonous Appointment Room 4 skip (is what i think you are refering to) - should work on PlayStation®2™.
To achive it on PS®3™ you need to press home button to make game lag, i have never played this game on PS®2™ so i don't know how it works on that console, but im pretty sure it should work the same.
For more info, join LEGO® Batman®: The videogame™ Discord server at https://discord.com/invite/cFtEhujMY2

So, is it ok to use 3-2 freeze glitch and nobody used it for a long time or is it just unacceptable thing to do in run?

so you can't submit runs that will be runned on normal PSP?

I do not know what advice i can give you. The best - in my opinion - will be starting game with order:
1-1
3-1
3-2
3-3
It will take you some time to do it to 3-3 but if you will do it 1-1, 1-2 etc. 2-1 2-2 etc. And then 3-1 3-2. etc. it will be end of the run so if you will do not get this good it will waste your 1-2 hours - I don't know how many time is it to speedrun on PSP. You can do it like3-1, 3-2, 3-3 but at the start of 1-1 you can go to Batcave. If you running Replay Story (not story) you can just start from 3-3 and then do what you want. You can also practice 3-3 boss fight to be the best 3-3 boss fight PSP player if you want to. I would maybe have LEGO Batman: The videogame on PSP so maybe i will give you some advice. I'm not a PSP player so i can not give you advice to how to play good on PSP, i can only give you some tips how to not waste 1-2 hours (again - I do not know how much time is speedrunning LEGO Batman: The videogame on PSP is).