Theoretically nobody can use the Royal Arms without a warrant; but in practice a lot of people do and always have, and provided they don't do it too blatantly in public they aren't likely to be pursued for it. Friendly and fraternal societies often used patriotic emblems on their artefacts - a crown, the Garter, the lion and unicorn.
I'm afraid old know all doesn't know anything! That is certainly *not* the badge of the Grand Lodge of England. It isn't even remotely like it - click on the link below to see the genuine arms of UGLE.
It doesn't sound Masonic at all to me. The symbol of the rayed eye isn't unique to the Masons; many friendly societies use it too, including the Oddfellows. As used by Oddfellows it symbolises charity, because "that alone can be true charity which is omniscient, and which can penetrate below the outward show of things".
What does the "Oddfellow crest" on your frame look like? In the later 19th century there were literally dozens of different Oddfellow unities in England alone, and they all had different emblems, although certain symbols are common to most of them - particularly the "heart on open hand", though Faith, Hope and Charity, the terrestrial globe, the lamb and flag etc. tend to crop up too. The full emblem of the Manchester Unity of Oddfellows, the biggest unity of all, included the shield of the Royal Arms (as well as the shield of the arms of the City of Manchester). You can see it (rather small, I'm afraid) at the HistoryShelf link. However, the motto "Friendship, Love and Truth" is more associated with the American Independent Order of Oddfellows (IOOF); so without knowing what that "crest" looks like I can't say what body it belongs to.