As mentioned, BA is meticulous in checking and double-checking for this ... there is 0% chance of getting by unless you know the agent and they do you a huge favour (for which they will get in trouble later, when the tickets are audited, as BA does 100% at LHR). Sometimes I've had tickets issued as stopover when I was planning on transit but wasn't sure if it would become a stopover later ... most airlines are happy to take a stopover as a transit (as there is a refund involved rather than a charge, though the reissue fee would also apply) and I've never had a problem with stopover being used as a transit (of course this becomes expensive at LHR!).

And 2 hours for a OWE ticket reissue by BA at LHR is very optimistic, more like 4 hours minimum and they are quite likely to say "come back tomorrow" and invoke a BA 24 hour re-issue rule. So this is a high-risk way to travel (or rather, not travel). The current mess at LHR probably doesn't bode well for BA doing you any favours.