I have never found anything with a QR skewer through it to be completely immune to movement in fixed gear use.
In terms of wheel movement the following applies;
chain full link = 1/2"
chain half-link = 1/4"
One tooth on chainring or sprocket = 1/8"
to which if you add
Other means = 1/8"
means you can always keep your chain slack free (even with vertical dropouts) if you have two chainrings 1T apart and a half-link to hand, provided you can somehow get a further 1/8" of hub movement.
Some means of doing this include
-"mix and match chains"; combine lengths of new and slightly (about 0.5%) worn chains to get the correct total length.
- file the dropouts slightly
- file the axle slightly, where it passes through the dropouts.
A combination of the last two is potentially better than you might expect. It won't hurt any frame if the vertical rear dropouts are taken from ~10.2mm to 11.2mm (eg by filing a small notch in the back of the slots with a round file). Once this is done an axle that is 8.0mm wide will move 1/8" within the dropouts; perfect! Except axles are often 3/8" (9.525"nominal, actually about 9.4mm across the thread tops) wide. Here's the clever bit; provided you are prepared to keep track of/adjust the axle 'timing' (which might require that you hold the axle with a spanner whilst tightening the first track nut, and are prepared to have the axle face forwards instead of backwards as the chain wears) you only need have a small flat on one side of the axle.
How small is small? Well, in a 10.2mm dropout the flat needs to be ~1.3mm deep but in an 11.2mm dropout the flat needs to be only 0.7mm deep to provide enough movement. This is about the same amount required to locally remove the protuberant parts of the thread on a 3/8" axle. A little more removal is required for a nominally 10mm axle.
This won't weaken it significantly. IME 3/8" and/or 10mm steel axles are almost a consumable in fixed gear bikes anyway, so this means that you can run relatively inexpensive parts and get your fixed gear, er, fix that way.
HTH
cheers