Can someone be in ESA Support group and still claim underlying entitlement to Carer’s allowance? 

As long as he has ESA(c), it will supress payment of CA.  The effect of that will be to generate a Carer Premium without affecting the couple-rate SDP.  You get the Carer Premium if entitled to CA, but only lose the SDP if CA is actually paid.

I understand about the carer’s premium, but might his being classed as a carer prejudice whether the DWP consider him entitled to his ESA or PIP?  I know it doesn’t matter in PC at the moment and didn’t used to make a difference with ESA, but just not sure now with all the changes.

The DWP has often taken   this line over the years .

I have attached an extract from a submission of mine answering that point

Adapt it if you need to at any time

It is a totally standard component of the welf toolkit.  If the couple’s care needs are compatible they can care for each other while needing care themselves, and given the overlap between the PIP and ESA activities and descriptors it’s not a huge reach to say that having LCWRA is also compatible with being a carer.  There are loads of cases like this.

If you are old enough to remember the 1990s docusoap Paddington Green, there was a couple in that show who would have been a perfect illustration of this point - one blind and one wheelchair user.

Used to manage staff who dealt with nothing but benefit take-up for older people. This was back in the day when full on genuine backdating was a thing. Was as described above absolutely routine to have couples with couples rate SDP and CP.  The gains were delicious 😊