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If this was my situation I'd now check again with BT Mobile again explaining Vodafone Uk have confirmed the PAC Code is correct.
Sounds like your stuck in the middle.
Is it Live Chat or Customer service on 191 providing the PAC Code.
It's not Ofcom but rather the Communications Omsbudsman that a person would engage with however the timeframe allowed for Vodafone to remedy raised complaints with customer services is 8 weeks before the Omsbudsman would become involved unless Vodafone agreed to a deadlock letter.
Alternatively, The Vodafone Community forum is monitored by Vodafone U.K. There are members of the Vodafone Tech Team who are available to respond and Help where they can. They have the Vodafone logo next to their name so a person knows they are officially Vodafone.
All posts are read by them. Hang in there and I'm sure they'll be in touch.
To be frank your current issue is rare to me and I've only come across this a few times before.
The problem is we here are both customers who understandably do not have any access to either Networks systems and what both sets of customer service frontline agents are seeing.
It could be either side at error.
If you're porting your number to BT Mobile, are you entering the PAC code online via My BT? if it's throwing up an error then the PAC is wrong as the systems on all networks check the PAC's validity before allowing you to proceed.
It would perhaps be worth asking Vodafone if there's any way of cancelling that PAC code so a new one can be issued as otherwise, the only other option (and it's a poor one) is wait the 30 days for it to expire before requesting a new one.
I wish you luck...
Perhaps someone at Vodafone would like to intervene on this thread and perhaps look into this issue as two days later, they are no further forward.
Perhaps it'd be worth Vodafone checking they've generated the right PAC code. I could imagine this being down to someone generating a PAC and noting it incorrectly on your account then every subsequent time you've called and checked, the agent's reading from the notes.
It'd be great if someone took ownership of this and contacted the team who are responsible for generating PAC codes and taking it from there.
Failing that, you're going to have to raise a complaint as this problem is stopping you transferring your number and whilst this is going on, you're paying for two separate services. Really poor.