This was kind of discussed here.
https://tropicalfruitforum.com/index.php?topic=48658.0To add to that, It has often been quoted that the Red Finger Lime is Citrus australasia var sanguine from Mt Tamborine, ( cool area )
however Red Finger Limes are found throughout the natural range of Finger Limes.
However these are still highly variable plants, some are from hotter areas and some from cool highlands.
Some selections might be affected differently by climate, i.e. flowering or fruit set.
The development of red pigmentation of some varieties seems to be climate affected.
It is possible that a particular selection of Red Fingerlime would perform poorly in some areas, but it is not a blanket rule for Red types as far as I know.
I think TFF member Mike from Cairns had Red Fingerlimes fruiting there, in the tropics.
As there are less Finger Lime selections available in USA, maybe see what happens, or grow it bigger to produce more flowers.
Not sure how different rootstocks might influence performance in hot climates ?