Looking at the GRO indexes, I can't see any Clark(e) births at all with mmn Beaumont in the Bradford registration district between 1890 and 1910, though there are a few in other districts not too far away.
So, are all the records you have consistent about his date/year of birth?
And, are any of the marriage witnesses possible siblings (or other relations)?
Also (ShaunJ might already have found this) are there any clues about his parents in his service record, if that is him?
Lastly, what do you know about Thomas's occupation - it might just help in tracking him down.
It would be quite unusual at that date for a birth not to be registered, so not being able to find it suggests an anomaly in the date, place or name. I did try looking for a birth under Beaumont, and there were a few of them, but more research would be needed to see if any ended with William Clarke as a stepfather, say.