Have a look at this.
https://www.forums.motester.co.uk/forum3/4007-1.html
Even Mot testers, it seems cannot agree.
But I would imagine that if your car can ONLY run on LPG as there is no petrol tank etc then there is no doubt you will not need an CAT. But if the vehicle can run on both then it'll have to have cats, for when it runs on petrol.
I believe MOT testers will ask you what you want it to be tested on, and my converted vehicle has, according to what I have seen on the Gov.uk MOT test part of the site, been tested on one, some times, then the other at other times.
If you always take your vehicles to the same place, like i do, and they are reasonable, then I'd ask them what they would do and would be looking for, as it seems to vary from MOT test station to test station.
At the end of the day, it seems as confusing to them as it is to us!