Sorry to keep labouring this point, but if your DH is not on the tenancy either as a tenant (like you) or a permitted occupier, he has no legal right to live in the property.
This is a legal minefield for LLs and without their express permission (in writing) your DH should NOT move in.
If you were a tenant of mine and I found out that another adult (except a child of yours that had become an adult in the life of the tenancy) was living in my property without my knowledge and the correct legalities then I am sorry but I would not renew that tenancy at the end of the term.
When a LL runs checks on their tenants, it is to ensure that they know who is living in their property and your LL has obviously decided that you are just the kind of tenant that they want to rent to.
Your LL has not carried out the same checks on your DH and I expect that part of the (ridiculous) £350 they want to put him on the tenancy is to carry out some basic checks.
TBH if I were you LL and you came to me with this situation I would offer to run the checks on your DH and decide whether I would want to rent to him.
If I did then I would probably add him on the AST for £25 (NOT £350) because of your honesty and that is what it would cost me to do that.
If I wouldn't rent to him alone then I am sorry but I don't want him living their with you wither and it would be your choice to either rent alone for you and your daughter or I would not give you the tenancy and rent to someone else.
If I found out later that your DH was living there, not just staying over once or twice a week, then at the next renewal, you would be getting a Section 21.
I don't know what other people would do but I do know that a lot of LLs have this way of dealing with non-permitted occupiers.