The frame does not have dedicated dropper post holes for routing the cable internally, as you have correctly noted. Here are the options that I can imagine in this situation.
Attempt to draw the cable via the bottom bracket via existing holes. It is hard to tell if you could pull this off, without seeing close-up shots of the area. I've never seen a "blind" seat tube, there is always an opening to the BB to at least drain water. But it does not mean that there is space to drag a cable. Besides, the sharp angle at which such cable would bend when transitioning to the downtube might cause friction in it compromising the action of the dropper post.
Drill a new hole in the seat tube to get the cable out of the seat tube, and route it along the down tube. All associated risks that come with such unauthorized frame modification are known.
Find a dropper post with external routing. This obviates the need to fish the cable out of the seat tube, as everything is routed externally along the top tube.
Find a wireless dropper post. This solves all the issues with the cable. I used a Magura's wireless post on such a frame that did not have provisions for internal dropper routing. The culprit here is of course that it is more expensive than rather affordable Brand-X seatposts. RockShox's wireless posts are even more expensive. The biggest issue I see is that your frame has 27.2 mm for the seatpost diameter, which is not the most popular size, and I do not know if any wireless dropper posts in such size exist.
In my opinion, the third option, an externally routed seatpost is the most reasonable one, especially for seat posts with the 27.2 mm diameter.
