Free Traffic Legal Advice

Is there anything I can do?

Yes,
Have a read of
https://www.ftla.uk/civil-penalty-charge-notices-(councils-tfl-and-so-on)/read-this-first-before-posting-your-case!-this-section-is-for-council-tfl-dartme/

and post up here all sides of the PCN. Redact only your name and address, leave everything else in.
I find using the BBCodes at https://imgbb.com/ works well.

Bristol can take a week or two  to catch up.

That said.

You should be able to ask them if there are any outstanding PCNs against your reg.mark.

Then, the PCN will go to the address printed on your V5C (is the vehicle yours or leased?). Don't assume the address is correct, check. (If the vehicle is leased come back here for further advice.)
 When you get the PCN, please post all sides here, redacting only name and address.

6 days is a ridiculously short period of time for payment

London gives 3 days  - IIRC it used to be by midnight the following day.

Anyway, is you V5C correct?

We must see the PCN when you get it, because a 2018 TPT adjudication case involving DART, under the same Road User Charging regulations as Bristol, decided that it is unlawful to demand the toll fee as part of a PCN. It should be noted that Transport for London ULEZ PCNs do not include any unpaid tolls, payment of the PCN concludes the matter. The London ULEZ is under separate regulations, but the content of PCNs definition is the same in both regulations

Is there anything I can do?

@bigwillie1979 pay the charge now, for the earliest date that you are able to. The tribunal held in the case of Luke Moran v Secretary of State for Transport (IA01249-1803, 13 June 2018) that an authority cannot hold a charge as an indefinite credit and they have to apply it to outstanding crossings.

If you get a PCN, you can simply give them the payment reference, explain that you didn't enter the zone on the date you actually paid for and ask them to reallocate the payment to the correct date.

If they refuse, we will help you with the tribunal appeal. The key issue is that the daily charge needs to be paid before the PCN is issued, so for the sake of £9 I'd suggest you do this tonight.