No, running it two or three times for every page isn't prohibitive. In fact, one of the arguments for using modern general-purpose multimodal models for historical HTR is that it is cheaper and faster than Transkribus.

What you can do is for instance to ask one model for a transcription, and ask a second model to compare the transcription to the image and correct any errors it finds. You actually have a lot of budget to try things like these if the alternative is to fine-tune your own model.