Bright children with difficulties can be comprehensively failed in the wrong school. A school that manages the behaviours but drops the academic expectations too low, is not what you want to aspire to for a 12 yr old, and that can be a risk if selecting a state school predominantly for its SEN support. (my impression based on reading threads on here)
For the OPs son, it is early days in terms of settling in secondary, given all the disruption at present. I agree that if PDA is in the picture it does need a very different approach to handling it compared with the majority of children. And at secondary it only takes one teacher "not on board" to be a problem :(
Why don't you try to arrange a meeting with the senco to discuss where things are at, and what you both expect to be achieved working together with the school. Making sure you're both on the same page and so on. I'd hope constructive plans can be put in place without / while waiting for a specific diagnosis.
One thing we had - in prep, so earlier - was plans for referral made, then postponed because things seemed to have improved. Then some months later the same discussion started again ... That was with DS, and he matured enough from the beginning of yr 7 to the end of yr 8 that we didn't need to mention his "ASD traits" when he transferred to senior school. (full diagnosis not pursued)
I hope some others with PDA experience can post here for you.