My first question would be, where about are you based?? If it was Scotland, I would suggest that you or someone like me could check that birth certificate, at Scotlands People Centre in Edinburgh ( and other places) - though because of covid, that might be a wee while!! They have been shut since lockdown or before. If you were somewhere further afield, then I would certainly offer to check at the first opportunity.
If your dad gets tested, I would suggest Ancestry, they have by far the biggest database, and you can then download the results, and upload to other companies, though not I think, unfortunately, 23&Me.
For more help on finding who the father(s) might be, it would be worth while checking the poorhouse records, and other local records in which poor families applied for relief, in times before the establishment of the NHS etc. It is surprising what details families will give away, if it means the difference between getting help, or not. I live in Lanarkshire, and can check these records at local historical archives, and indeed on line I think!
I should have said above, I think an autosomal test ( at Ancestry) is your Dad's best way forward. especially if your family didnt move about too much.
For example, my mum's 2 sides were both irish, but in scotland for a long time, staying in the one village area for >75 years. I live quite close too. The result is that I know all the family names common in the area, and if I get an unknown match, who matches with my many known matches in that part of the family, then I can be fairly sure where they fit in. Similarly, if they have a known surname, or their tree contains known surnames from the area, to me, again I can be pretty definite that they relate via that part of the family!! Though you need to back it up with supporting records, of course!
Hope that helps a wee bit!