The recommended duration of cyclical progestogen is 10 days or more.
Two pumps of gel is also not a lot, so it's not as if you have been on high dose estrogen all this time either.
Bleeding that occurs within the first 6 months of a change in hormone therapy is not normally considered a red flag and doesn't warrant a 2ww referral.
However they may be a bit twitchy about the fact that you are on cyclical progesterone as well, as guidelines generally recommend continuous use over the long term.
(I personally don't agree with blanket recommendations that prioritise avoidance of a single disease with low prevalence and high cure rates at the expense of multiple other clinical outcomes eg breast cancer risk, quality of life, cardiovascular disease and cognitive health, all of which show favourable outcomes with cyclical vs continuous, however those are the guidelines they are working with.)
At least you will be scanned quickly then I would recommend another discussion with your private consultant, informed by the lining thickness on your scan.
It is also a good idea if on individualised treatment with either a lower dose or shorter duration of progesterone than the NHS recommends, to have an annual pelvic scan privately to monitor your endometrium, that way you know you are covering all bases.