This one actually seems pretty straightforward but is hard to prove without archival material.

I strongly suspect that the RFC is connected with the local gentry, specifically the Meades of Clanwilliam. The Meades were made Earls of Clanwilliam in 1776 and from the 1770s and 1780s held estates in Ballygriffin in the Barony of Clanwilliam (from where they derived their title). It looks like the Meades are the source of this double-headed eagle.

A heraldry blog has an article on the sale of bookplate allegedly belonging to the son of the third Earl of Clanwilliam, dated 1879. The bookplate features a double-headed eagle crest, the crest being the device (often but not always an animal) that goes above the actual coat of arms. The obvious thing to do is to see if this crest is generally associated with the Meades of Clanwilliam.

I checked Burke's Peerage - your standard guide to the British nobility of England, Scotland, and Ireland - for the Earls of Clanwilliam to see if it gave their heraldry, and indeed it does:

Arms — Gu., a chevron, erm., between three trefoils, slipped, arg. Crest — An eagle, displayed, with two heads, sa., armed or. Supporters — Dexter, an eagle, close, sa., sinister, a falcon, close ppr., beaked and legged or, each collared and chained of the last. Motto - Toujours prest.

I've emphasised the important part above and (to translate) it basically says:

Crest: a black, two-headed eagle with wings spread and with golden talons.

I checked the 1878 edition to be absolutely sure it was around the time Clanwilliam RFC was established (even though these things almost never change very much anyway).

The problem here, however, is proving a link between the two. It would seem to me to be a safe bet that the Meades were patrons of the club at some point, but without knowing the sponsors or the membership of the club it's hard to prove. I know personally that clubs tend to 'borrow' from the arms of their town and county and from those of noble patrons but hopefully someone with more experience in the history of sport can chip in.

I would be curious myself to learn where the double-headed eagle comes from with the Meades . The official centenary history does at least indicate that the team was likely set up among soldiers at the local barracks but gives no hints as to the origin of the club badge. As you seem to have already discovered, I don't believe any British army formations used a double-headed eagle insignia at that time. It does have precedence in Irish heraldry: for example the Browns of Galway are credited with a black, double-headed eagle on gold (or, rarely, silver):

Arms. Or. an eagle displayed, with two heads, sable.

(Incidentally Burke's Peerage indicates that the first notable Meade, John Meade, was related to the Sarsfields and Butlers, both very prominent families in Munster during the seventeenth century (and much earlier, for the Butlers), but neither ever used double-headed eagle devices.)

Hopefully this helps show the possibility of a relationship (at at least one time, anyway) between the RFC and the local gentry - and also that it is helpful to think of the club in terms of the local community first. The first match - according to the centenary history - was between Clanwilliam and the local Abbey School, after all. However, I think it would be impossible to actually conclusively prove or disprove this conjecture without having access to the archives of either the RFC or the Clanwilliam Meades.

NUI Galway, 'Estate: Meade', Landed Estates Database: online

Stephen J F Plowman, 'Bookplate: The Hon. Robert Henry Meade', The Heraldry Blog (11 Feb 2017), online

Bernard Burke, Burke's genealogical and heraldic history of the peerage... (1878), p. 245,

James Hardiman, The History of the Town and County of the Town of Galway (1820), p. 10.

Denis Marnane, Clanwilliam Football Club 1879-1979 (1979).