I would split the original invoice. The UC payments are there to support you because of your financial circumstances. I don’t think it’s unreasonable of you to ensure that you benefit from that. These kinds of benefits are calculated on your income. No point in you trying to pay more if you can’t afford it!
I personally think it depends how much UC are paying you back. For example if your fees are 1000, UC pay you 600 and you’re taking 500 from your partner that wouldn’t seem fair. I’ve no experience with UC but would you not need to update your application with your share of the childcare to keep it accurate? I’d be worried that I’m lying on the application if I’m saying childcare is X but you’re having half of that given by ex.
I would just get him to give you half of the nursery fees. UC won’t cover all of it.