Some time has passed on this thread, but I’ll chip in with some practical information.
If your DH decides to search for his father, there are non-profit agencies, e.g Salvation Army, that can help people & part of that process is some preparatory discussion/counselling before the search begins. The opportunity to talk this through with someone experienced in these matters may help your DH considerably.
Critical to any search is useful reliable data. Your MiL is unlikely to shift her stance on this, so your DH as an adult is free to go round her. First stop is his aunt - what can she tell him about his father, e.g name (Bobby Bloggs),approximate age (early 20s in 1980s) where he lived (within a few streets) , any known other relatives (mum Gertie, dad Balthazar aka Bert, sis Delilah etc ). Build up a rough portrait.
I did a search for someone in roughly the same situation & was successful. I searched online for electoral rolls for the known area in 1980s to determine where they were held, then booked an appointment to view the records. This took me to the British Library. Trawling through the records, I found the exact address of the family involved. So I now had a set of people over age of 18. From this, I searched the BMD registers for births, marriages & deaths for those recorded. This allowed me to cast forward to current day with full names & ages of those people & their children. I ordered copies of a couple marriage certificates to double check my facts. Checking online dbs like 123.com gave me current addresses.
I also checked Facebook for names of youngest members of the family. I found the whole family online including photos. Privacy settings (!).
Not difficult if your DH is organised & determined.
Under no circumstances, would I contact the people I found directly. I gave the initial sets of facts to the person searching to be shared with an agency. For me, finding what I could was solving a puzzle & I was careful not to get over involved or go too far in a highly sensitive situation.
Giving a few facts to an agency who know a) what to do & b) how to approach people in this situations & c) manage the outcome would be the wisest course.