I've been in a similar position, although my early scan was already booked for 10 + 5 as my NHS scan wasn't happening until 13 + 6 and we wanted to know before a particular date.
I started spotting/bleeding a few days before the scan. I spoke to my midwife and she said that the EPU wouldn't see me any sooner than the private scan I already had booked and to go ahead with that.
I found out at the private scan that I had miscarried at 6 weeks ish. Handing over my card to pay the final balance on the card in front of a full waiting room of excited people looking at scan pictures or waiting for scans was a pretty shitty feeling to be honest. Although the woman scanning me had herself been lovely she was paid to do the scan only and there wasn't any kind of aftercare or any process in place at the scan place (a big national chain) for you to have somewhere quiet to go afterwards.
The final shit icing on the shit cake was a few hours later I got an automated text message from the company saying that they hoped I had enjoyed meeting our baby today and would I like to book in for a gender scan at 16 weeks.
One angry and emotional email later I got a very quick, personal and nice apology from the company, but honestly it was altogether not a very nice experience made worse by the private scan.
On the other hand though when I did actually physically miscarry about 24 hours later I was very glad to have had a bit of warning that it was going to happen.
I resisted temptation to have any private scans with my subsequent two pregnancies (DC1 currently 15mo and DC2 due in 6 weeks
) but that's mainly because the angry email I sent the company at the time effectively said I would never darken their doorstep again, and I was too principled to go back on that 