Not from Bristol, although there are advances routed "ATW GWR NT & TPE" available between Newport and Middlesbrough, in theory. I say in theory because there is an error in the coding of that route in the data used by electronic booking engines, which has the effect of requiring that journeys go via London, which then makes the TOC restrictions impossible to achieve.
Booking engines powered by the Silverrail IPTIS or Atos WebTIS journey planners, which means most of them, i.e. National Rail Enquiries, Trainline, all Trainline-powered sites and all WebTIS-powered sites won't offer these advances for a journey from Newport to Middlesbrough. However, booking engines powered by the Fujitsu RIBS journey planner (e.g. Raileasy or Train Genius) seem to be interpreting the via London requirement wrongly and as a result allowing these advances, so that would be your best bet for getting one of them.
And no, Bristol to Middlesbrough isn't permitted that way (but Newport to Middlesbrough is, as mentioned).