I put Noahs moses basket in his cot, I never had any problems with him getting distressed, some bubs may get upset when transfering over from one to the other as they prefer the security of a smaller space, but you can always try swadling them in the cot, on the other had other babies are the other way round and dont like feeling restricted. Their is certianly no problem with you poping him in a cot at this stage, it might just be a bit of trial and error to see what way they like it, lots of space or swadling.
If you would rather keep him in the moses basket cause they are in a room with you, persoanlly I have never heard of a baby tipping itself out of a moses basket, but if you have them in a stand at the mo, perhaps you could put it on the floor instead to be on the same side, or just as your HVs advice.