Could this be one of those "irregular marriages" I've read about and if so, would using the other name be recognised for any legal purpose?
Small can of worms here.
An irregular marriage was (usually) contracted by the couple declaring themselves to be married in the presence of two witnesses. This was a perfectly legal marriage in the eyes of the law, but not in the eyes of the kirk, which would, given half a chance, admonish and fine the offenders.
After the start of civil registration in 1855 and until the law was changed in ?1939 you could still get married by declaration, but you then had to get the marriage registered by going to the Sheriff with your witnesses and making a statement that you had married this way, backed up by the witnesses. You then got a warrant that you took to the Registrar authorising him to make a record of it in his register of marriages.
So the expression 'irregular marriage' used by the kirk ceased to have any real meaning after 1855, when marriages that the kirk would have regarded as 'irregular' were entered in the civil registers. No doubt the kirk continued to fulminate against such marriages, but in vain.
Your Elizabeth's dates are well after 1855, so I very much doubt that an 'irregular' marriage comes into it at all, unless she and John Bruce declared themselves married in front of witnesses and then neglected to get a sheriff's warrant and have the marriage properly registered.
As for legal recognition of the second surname, I doubt it.
But all this is generalisation, and I'm sure people will be able to come up with specific cases to contradict the foregoing.