Here are some observations and opinions of mine regarding Bermuda's 'Papa Gontier', "Mrs. Heggie's Red Tea", 'Beauté Inconstante' and 'Papa Gontier' having considered the photos and references and having only grown the 'Papa Gontier' that we have in Australia.

Firstly, if the bushes of "Mrs. Heggie's Red Tea" shown in the photos of MargaretFurness (2Jan11, id 165778) and in particular JenLight (4Apr16, id279335, the mother plant) are typical, then I don't believe they could be the same variety as BermPG as photographed by mmanners. Even allowing for different cultural situations the habits are too different, and there are other differences as well.

One of the big questions retained in my mind as to the whether the Australian PG is the correct variety, is that it doesn't (or extremely rarely) produces hips. This doesn't correlate with the records that show that PG was used as seed parent on numerous occasions such as with 'Lady Hillingdon' and 'Comtesse Emmeline de Guigne'. There is even a botanical drawing of a PG hip. So it is interesting to see that Malcolm's BPG produces hips, even if they aren't as ovoid as the known drawing.
Also, as we know PG was used extensively as a greenhouse florist rose in the late 19th century, particularly in America. The style and demand was for long stems. This is something I've never observed AusPG to display and even imagining what could be achieved by forcing and feeding, it's hard to see AusPG's short usually nodding somewhat kinky stems being a raging success as an early florist rose. That's where BPG becomes interesting because mmanners shows a large bush with long straightish stems very apparent at the apex. This corresponds to the early descriptions of PG such as the Riviera reference and other florists references.
The just-opening buds in Malcolm's photos also matches the early references to being filbert-like and oddly truncated at the ends. Being compared to a filbert is very much open to interpretation, but most filbert's have a somewhat bulbous base, constrict a little in the middle, then flare out at the end. Again this is suggested to some degree in the Rosenzeitung illustration of PG.
Another big concern I have about the AusPG is it's apparent lack of similarity to PG's known sport 'Rainbow', in particular the habit and leaf shape. Even if we are just assuming that the roses grown as 'Rainbow' are correct (or could at least be 'Improved Rainbow') the foliage, structure of the bush and the flower form bear more resemblance to BPG than they do to AusPG, in my opinion.
Add to this that the flowers of BPG often show flecks or partial stripes, this suggests to me that BPG is more likely to produce a fixed stripe sport than AusPG would be (well, this is speculation but it seems logical).
Additionally, the foliage of PG shown in early catalogue photos suggest a somewhat broader or rounder leaflet than AusPG which I find is very classically and slenderly acuminate. The broader leaf shape is seen on photos here of 'Rainbow' and also BPG. Also, Malcolm's beautiful photos of BPG are taken in midwinter in Florida and show a healthy-foliaged winter flowering rose, which PG was of course known to be,although AusPG does this too of course.
Couple all this with the fact that BPG has a (at least verbal) provenance as coming from a bunch of florist roses of PG, I think that Bermuda Papa Gontier should be seriously considered as the strongest possibility for the real identity of Papa Gontier, and that the Australian Papa Gontier is some other cultivar, in my opinion.

Regarding Mrs Heggie's Red Tea being Beauté Inconstante, I have less to say. All I would point out is that the European version of BI doesn't appear to be the same as MrsHRT just looking at the photos here. Additionally, looking at the Thomas photo and the Journal des Roses illustration, the most notable similarity between these two is the rolling of the edges of the petals which is even present in the opening bud. That might be a distinctive way of helping to confirm this rose. The colour between these two depictions is similar, but in such a variable rose I guess that is nothing conclusive.
Also I'd like to know from those who grow MrsHRT if it's very prone to mildew like references imply BI was.
Anyway, these are just my considered observations. Feel free to agree or disagree. Discussion on this site is a great thing.