I’m not an accountant but my gut feeling from what I’ve read of the HMRC guidance on disbursements is that this isn’t a disbursement. Something is only a disbursement if it’s basically being supplied to the client but you’re paying for it, and the client knows that this is the arrangement and it’s not something being supplied directly by you. If it’s something that has been supplied to you and then you’re selling it on to the client then it’s not a disbursement but a re-charge, and you need to charge VAT on your onward sale.
If your client is themselves VAT registered then it makes no difference to them so you might as well take the easy route of doing the reverse charge yourself and then re-selling the service to your client at 20% (which they can reclaim).
If your client is not VAT registered, then if the supplier were supplying the client directly they would have added their own VAT to their invoice rather than supplying on a reverse charge basis (either their home country VAT or 20% UK VAT under MOSS depending exactly what kind of service we’re talking about). The fact that the supplier has invoiced you as reverse charge suggests that they think the supply is to you, not to your client, and so it doesn’t feel like it should be a disbursement, so again you end up having to charge your own 20% VAT on the re-charge.