Single speed and a chain tensioner will work no problem
In general, fixed gear in these situations is not a good idea.
It could work if you use a half-link chain and choose your rear cog and front chainring so the size works perfect with the chainstay length. But this will fail over time as the chain wears out and the fit becomes loose.
The issue with chain tensioners and fixed gear is that when you use a spring tensioner and apply forward pedalling pressure, there is no problem, but when you apply backwards pressure, the chain will becaulse loose and will most probably come off. When you stop the forward pressure, the chain will become tight at the bottom (yellow arrow on the picture) and loose at the top.
If you try a non spring tensioner, such as the DMR STS, the forces you generate are stronger than the bolts on the tensioner, so you will be constantly re-adjusting the tensioner.
In my experience, as I already tried this, you would have to fine a chain tensioner which works on the bottom of the chainstay when you pedal forward and one on the top when you put pressure backwards. And non spring tensioner just come loose very easily. I suggest you go down the single speed route and not the fixed gear route.
Good luck and well done for fixing up that frame.