Welcome to the forums James
The simple answer is that it IS possible. However, in the circumstances you describe, unless your employee has a raft of other clients IMHO it is unlikely that they will escape being classed as an employee by HMRC
Why not simply vary their existing contract to cover this need (ad hoc overtime upon request) and charge the extra cost to the other department?

I had this query come up before and my position was that this was a very bad idea. Not only does it introduce some very blurry lines in terms of status but also raises potential conflicts of interest if the employee is better paid for one aspect of work over the other.

In my case, we resolved it with a temporary contract change. We allocated a fixed proportion of the employee's hours to the loan department and they allocated budget towards an honorarium to reflect the project work time. IIRC, the employee then worked three days pw in their regular role and two days pw in the project role for four months before the arrangement came to an end and he returned to his old job, full time.

In reply to Ray:

Thank you Ray for your advice. I was thinking along the same lines and am reassured by your response.

In reply to Robey:

Thank you very much Robey for another good suggestion on how these cases can be flexibly managed. Tricky as I'm finding it hard to prove why this is not a good idea, but I'm reassured you are thinking along the same lines as me.

In reply to James:

It's one of those things that, as long as everyone acts honourably and honestly and gets on well, would be absolutely fine.

But the moment the relationship goes in any way sour, you find yourself up against a barrel wishing you'd taken a more structured approach from the start.

I am amazed, now that I'm out of th risk-averse public sector and working in a commercial role, by how much serious, high-value business is still conducted on the basis of a handshake and a nod.

In reply to Robey:

Re nodding things through and one's word being one's bond, it amazed me how literally millions of pounds could hang on all this. In the particular engineering construction business I once worked in, the ‘civils’ - groundwork’s and foundations etc - of any major job typically were worth tens or hundreds of millions of pounds but the nature of the beast was that no one could be sure exactly how costly they’d in reality be .until the ground involved actually got dug into and the necessary concrete poured etc. etc.

So, all the voluminous estimating and tendering and subsequent contractual documentation involved often got totally disregarded in favour of the outcome of a chat over a cup of tea and a nod and a wink and a handshake between the MD of the main civil engineering contractors and our own MD based on little more than their personal relationship and absolute trust in one another and the principle of ‘some you win and some you lose’ - had such as this not been possible, total financial mayhem and costly arguments and litigation would have happened endlessly - as is the norm as Robey observes for the public sector.