Probably have to soak 'em pretty good to get right into & between all the individual strands that belts are made out of.
That little label would either have to be removed & re-sewn on after, or taped off. That has to be visible in order to conform to DOT rules.
If you're ever in an accident, (hope not) & an insurance guy sees that the belts have been modified, (dyeing would be considered a mod) then you may be S.O.L. as far as injury claims go. Insurance adjusters don't need much of a reason to say 'NO'.
Seat belts are easy to make, (for me, anyhow!) but I absolutely won't. No way! That would put me on the hook for any injuries incurred, if they failed. Big 'IF', but I'm not going to chance that.
Aftermarket belts for any kind of race car are not worth what they charge for 'em, as far as materials or labor involved to produce them. The cost is due to that label. That shows that the manufacturer has met a standard set out by D.O.T., & MUST be purchased from D.O.T.
In order for a manufacturer to buy that label, they have to prove compliance to D.O.T.'s rules, & carry the liability insurance that goes along with that. That's where the big bucks are required. That liability insurance expense is stupid huge!