Issue here is they saying: "we are to big company, it is our standard contract, we will not change contract just for you ... at least that is what their HR is saying"
If you ask a prospective employer to remove any clauses that say "Any work done during employment is ours" then why should they bother? They put this clause in to a) avoid being the funder for work that doesn't benefit them and b) Ensure you don't steal other company resources.
If you suggested "Can we please add a clause that states any work done outside of contracted hours using resources not directly owned by the employer shall remain the property of the employee" that would seem much more reasonable.