A baby at the breast will of course trigger milk production, but it is the continual emptying of a breast that is what stimulates continuous milk production, not a 10 minute breast feed that is then stopped.
Read up about the Feedback Inhibitor of Lactation.
Stopping a baby who is actively breast feeding in order to give them a bottle, and to then pump is nonsense.
Why stop a baby who is feeding and use a pump to get milk out when the baby was doing it anyway?!
A baby who is feeding well is FAR better at getting milk from a breast than a pump is and so effective breastfeeding with no supplementation is the best way to improve milk supply.
However, this is only appropriate if the baby is feeding correctly - I.e the lactation consultant is happy with the baby’s latch and is confident the baby is transferring milk well.
One thing to hear in mind though is that at this stage there is always the possibility that you will never be able to build up your milk supply to the point where you can exclusively breastfeed. The lactocyte cells that specifically make the milk close off if they are not required (usually associated with low milk supply) and once they have closed up they cannot be opened again so your supply will be dependent on how many functioning lactocytes you have.
I think the first two weeks post birth are the most important for priming the lactocytes to be able to provide a good and consistent milk supply so if there are initial difficulties within that window then it’s likely lactocyte cell death may have occurred which is why you are having continued supply issues.
This is something your lactation consultant can discuss with you in more more detail.
How much milk are you expressing? And in terms of increasing expressed milk volume it’s always better to express from both breasts within the same pumping session so a double breast pump would be recommended.
Finding find time triple feed (breast, bottle and pump) is extremely time consuming and being realistic, impossible to maintain long-term.
I think you should speak to your lactation consultant ion and have a really frank discussion with her about what you want and whether she thinks it is possible due to how many weeks post birth you are? Is it that you want to return to exclusive breast feeding or are you happy combination feeding?
In situations like yours, if you are happy to combination feed then I would let your baby breast feed for as long as she wants to as long as she is latched well and taking milk properly, and then offer her formula afterwards.