I've dug into this a wee bit more, and it really is rather muddled, even murky.
It seems clear that in the early 30's, when the IFA was calling up players from all over Ireland, the FAIFS had a policy of only selecting players from the Free State. Yet as Harry Chatton shows, even that policy was not "set in stone":
http://nifootball.blogspot.com/2006/...y-chatton.html
As to whether the FAIFS also had a policy during this period of declining to select Southern players who were also willing to play for the IFA, that may also be broadly true, as possibly illustrated e.g. by Alex Stevenson:
http://nifootball.blogspot.com/2007/...stevenson.html
Yet there are exceptions to that "rule", too. One notable such case was Jimmy Kelly, who debuted for the IFA in Nov.31 and went on to become a regular in the team, gaining 11 IFA caps in total until 1936. Yet he also was capped by the FAIFS in May 32 and Feb.34, gaining 3 more caps for them in 1936.
Curiously, he never played internationally for either team after that year, despite being still only in his mid-20's, yet he must still have been a very decent player (at Derry City), since he continued to represent the Irish League in what were then very prestigious games against the English and Scottish Leagues for some time (also the League of Ireland!).
http://nifootball.blogspot.com/2007/01/jimmy-kelly.html
Another such case was Paddy Moore, who made his debut for the FAIFS in 1931, gaining his 7th and final cap in 1936 (when he was still only 27, btw).
Yet in between, he played once for the IFA team, away to England in 1932:
http://nifootball.blogspot.com/2007/09/paddy-moore.html
I suspect both Associations may have been influenced by additional factors, so that apparently fixed "policies" may actually have been somewhat "fluid"!
One likely (ahem) seminal such factor was the fixture clash in 1931, when the IFA XI were at home to Wales on 22nd April, whereas the FAIFS XI were playing Spain in Barcelona four days later. I imagine that with travel logistics etc, it may not have been physically possible for any player who wanted, to play in both.
Eventually two Southern players who had previously represented the IFA, Tom Farquharson and Peter Kavanagh, opted for the FAIFS, but two more such players, Jimmy Dunne (as mentioned) and Harry Duggan, stuck with the IFA.
Farquharson had represented the IFA seven times from 1923-25. He debuted for the FAIFS in 1929, played again for them in 1930 and gained (only) two more FAIFS caps during 1931.
Kavanagh played once for the IFA in 1929, then twice for the FAIFS (inc.in Spain) in 1931. He was not capped subsequently by either, despite still being very young (b.1909), but that appears to be because after being a "boy wonder", his career nosedived soon after.
(Their cases indicate that the IFA probably did not discriminate against Southern players who also chose to play for the FAIFS, whether previously or concurrently, btw)
Meanwhile, Duggan who won 9 IFA caps between 1929-35, debuted for the FAIFS in 1927, wasn't capped again until May 35, got two more in 1936 then got his final (5th) cap for them in Nov.37, by which time he was 34.
As for Jimmy Dunne, he was capped 7 times by the IFA between 1928-32. In between, he was awarded his first FAIFS cap in May 1930, but was not picked again by them until 1936, which for so prolific a goalscorer, cannot have been a coincidence.
So both Duggan and Dunne may have "suffered" somewhat with the FAIFS for their opting for the IFA, but that may have been down to that specific choice they had made in April 1931, rather than a general policy of not picking any player who ever played for the IFA?
And another defining year in this story seems to have been 1936, when attitudes at the FAIFS hardened, so that they renamed themselves the FAI, and their team simply "Ireland" (vs Switzerland on St.Patrick's Day!). Further, they began selecting Northern players, as well as restoring Southern players who may previously have been overlooked due to their having represented the IFA. Such an attitude was presumably reinforced by constitutional change in 1937, and possibly also by a greater determination to qualify for the 1938 World Cup, following their initial attempt in 1934, and especially since the IFA were not taking part.
Of course, what the Associations wanted was not the full story - the players' attitudes will also have played some part. It might be expected that personal political preferences might have influenced some, especially when memories of Partition etc were still raw. Tom Farquharson was a fledgling activist in the War of Independence, eventually appearing to have been sent to Wales to keep him out of further trouble. Therefore one might expect his allegiances to have lain with the FAIFS.
Yet the aforementioned Jimmy Dunne, who chose the IFA over the FAIFS in 1931, actually came from a noted Republican family in Dublin, and was even interned by the Free State Government in the Civil War!
Of course, it may that then, as now, many footballers weren't much interested in politics, preferring instead just to play football. In which case the IFA might have been a more attractive proposition. For one thing, unlike the FAIFS, they regularly played England and Scotland, which were huge games, which would have been a prestigious "shop window" for ambitious young club players. They also possibly had more money than the FAIFS, which might have filtered down to the players in better arrangements and match fees etc. Also, there will inevitably have been less arduous travelling etc involved in IFA games (i.e. within the UK), whereas the FAI more frequently played on the Continent.
And the FAIFS's opponents will often not have been so "glamourous" as they would be today. For example, Netherlands football was amateur or maybe semi-pro in those days and not of a very high standard.
Plus, as mentioned earlier, the IFA did appear to make an effort with Southern players - possibly in order to make a point?
Finally, there is always the possibility of "local" factors. For instance, Southern players playing for Irish League clubs were, I think, paid better than in the LOI, so their clubs may have steered them towards the IFA. Ballybofey man Jimmy Kelly certainly played for a number of IL clubs, in particular Derry City. It might therefore have been simpler for him to play in Belfast than Dublin, or in GB than on the Continent.
As for those players who played in England, I've seen reference (Farquharson?) to pressure being put on them by the Football League not to represent the FAIFS, since they (FL) were in dispute with them (FAIFS) for a period.
And I'd guess the other three "Home" Associations backed the IFA were they could.
And finally, of course, personal considerations may have come into it. For example, players will occasionally have fallen foul of selectors and "Blazers" etc at one or other of the Associations. Or there may have been more venal factors, such as the Shamrock Rovers rumours you allude to - who knows?
It's certainly fertile ground for further proper research.