Do people with congenital analgesia feel cold?
Interesting question. Here what I think (I am not an expert in congenital analgesia so take it with a grain of salt).

The pain provoked by an high temperature is mainly transmitted via the receptor TRPV1 (a Ca2+-channel nocireceptor) while cold sensation is driven by TRPM8 and possibly, for noxious cold, via TRPA1 (two similar types of Ca2+-channels). Those receptors are known as the temperature sensors. As an anecdote, TRPV1 is what give you a pungency sensation when eating spices while TRPM8/TRPA1 give you the cooling sensation from mint.

Mechanical pain is transmitted via other receptors, including TRP channels, but this is not fully understood at the molecular level.

As far as I know, the encoding of pain, and more generally physiological sensations, in the brain are also not yet fully understood but it is very likely that the pathways taken by TRPV1 differs from the one taken by TRPM8/A1 which would suggest that it is possible that CIPA individuals experience high-temperature pain and cold differently at the molecular level and therefore might respond, to some degree, to one but not the other. This also probably applies to mechanical pain vs temperature-driven pain.

I would actually not be too surprised if CIVA people shiver even if not feeling any temperature sensations. Would be interesting to have the opinion or even better the observations of experts in this field.

P.S. I know this is not a definitive answer but I have not enough reputation to comment.