2. No Time Lords Means No Limit
That the Time Lords could provide someone with extra lives implies that they have control over the process of regeneration. That squares with what we saw in "The War Games", where they forced the Second Doctor to regenerate, but offered him his choice of new bodies. So perhaps if they have that level of control, perhaps the limit of thirteen lives is artificial. Perhaps it is something they imposed upon themselves to prevent over-population or something. If that's the case, it could be easily established that as there are no other Time Lords around, there's no longer an enforced limit, and the Doctor really can regenerate pretty much as often as he wants to. That removes the problem completely and, as long as the Time Lords stay gone, it remains a valid fix until the Doctor actually properly dies. Then there's the last work-around.