80% done!
My suspicion was true - the reach is MTB-like and requires short stem with flat bar. I'm definitely getting both brakes.
Will probably ride a bit on freewheel before committing to fixed gear.
As of now the bike is technically rideable - set on fixed gear allows you to break. But the interaction with the bike and position are poor. With shorter reach, comfortable lock on grips it will be very different. I will need better pedals and comfy saddle before covering any significant distance too.
I have a lighter front wheel awaiting service. The super narrow 25c tires are terrible. The fork would only take 27-28 max - I might stick to 25c for now as it's a new tire. For the rear I have semi-used 32 from gravel bike that would be great. The clearance at the back is not massive, but I think it will do if I set the wheel perfectly straight. I hope I won't need tensioners.