First, welcome to Flyertalk! It's not unusual for people to first join when something unpleasant happens but of course one can only hope things will get better!

A first thing to note is that with all airlines (not just BA), the investigation of suspected fraud etc is voluntarily kept quite obscure. There is little communication from the airline and no way of directly getting in touch with the people leading the investigation unless they get in touch with you themselves. There is a long thread on suspected fraud, account suspension etc which I would recommend you look up and check as it will give a tonne of useful background.

I think it is also important to note that in such cases, the devil is often in the details. I assume the following based on your description but would be good to know if any of it is incorrect:

1) Your housemate and you decided to go on a trip together. You chose to have him send you avios so you could book for both rather than each book your own ticket - is that correct (and why I guess? Is it because you have higher status and there would be benefits in your doing the booking for both?)

2) He transferred the avios to you through the "transfer avios" function on the website

3) You then decided to cancel the trip and the avios were refunded to you

4) You thus used the avios transfer function on the website again to transfer back to him/her the avios they had sent you in the first place.

5) Your housemate was not contacted by BAC but saw that the avios had not reached their account (or had reached it but were deducted and the account was frozen? (What happened what did he/she see?) so they contacted BAC to ask what was happening

6) He/she received the answer you quoted above without having been asked any question as to who the avios were coming from or why

7) You have not had any communication from BAC and your account is functionning normally.

If any of the above is incorrect or inaccurate it would be good to know because again, even small details which you may feel do not matter may do and make the advice you receive here mistaken/useless if it is based on the wrong assumptions.

I think if all is correct, the communication should come from your housemate, not from you. Your account (you) hasn't been affected, so your commmunication is likely to just confuse things or draw a blank for gdpr reasons.

Your housemate is the one who should explain, as simply as possible what has happened and why they feel that the accusation is mistaken and that the avios were in fact transferred consistently with the rules. I would stress that there was no payment at any stage, and that the transfer was merely to enable the two of you to have a single booking for the two of you travelling together with each passenger effectively paying for their ticket, and that conversely, when the trip was cancelled, that transfer of avios had not reaosn to be any more.

Good luck.