Many thanks for the advice everybody, especially @ojs
I'll need to get a BB with a longer axle.
According to FSA tech support, the Vero uses a 110.5 axle length for a 44mm chainline, and the Tempo requires a 118 axle length for a 44mm chainline.
I have done some road conversions to fixed gear as well, and find that 68x103mm bottom bracket works well for getting as close to perfect chain line as possible. Essentially, the shorter the spindle length (103mm in this case), the closer the chain gets brought in toward the bike, which is what you want when the chain ring pushes it out further than the cog ...
First, just that both say "FSA Road" doesn't mean that they are compatible.
The answer is to look up the documentation for the crankset. It's the first Google hit if you search with the name and then look for bottom bracket. It says 118mm (which is unusually long) and JIS. You need a bottom bracket with axle like this.
Edit: Looking at the bottom ...
The skipping may come progressively often because you apply progressively more power (higher gears). I assume you are trying the smaller cogs at similar speeds.
Chain length may be an issue, but most likely the issue is the cassette. If your old chain skipped a lot, it is almost sure that it abraded a lot of of material from the cassette, especially when ...
When you are on the largest chainring and largest sprocket the rear derailleur should look very extended. Some movement should still be possible. It should shift freely from second largest to largest sprocket and you should be able to e.g. push the chain to the chainstay.
On the small-small combination the derailleur shouldn’t be completely folded, i.e. ...
The cassette does not look worn out
A cassette that is worn does not look worn.
It'll skip anyway despite not looking worn.
I suspect your cassette is simply too worn.
(Interestingly, for chainrings the opposite is true: a chainring can look very visibly worn due to the non-even wearing at different crank orientations, and still work.)
The new chain barely ...