With that specific cabinet design I do not think that I would overly concern myself with removing the existing crossover. If you had to leave it in place that would be okay. There is nothing on that board that is high enough value or quality that make me want to reuse it. If it was my speaker and I were placing a new crossover in that cabinet, I'd start by just pulling the woofer and see what you can see. I wouldn't lose sleep over running new wire all the way to the coaxial. Replacing it would be great but If removing it was enough of a challenge I'd consider just grabbing onto the coaxial speaker wire where it exits its secondary internal enclosure and connect new wire at that location. Secondly pick a location to place the new crossover board that is physically accessible from the woofer opening. I'd consider placement on the floor of cabinet depending on how long the port tube is, adhered to the exterior back wall of the coaxial's internal closure, or against the upper portion of that same internal dividing wall that prevents you from accessing the existing board. The cabinet is large enough that I'm confident that if you can fit it through the woofer opening than you'll be able to find a wall to mount it to. If needed, you could split the crossover to two boards and put the woofer circuit on a different board than that of the coaxial. That might be a good option for this design.
It's a great speaker and I think it would be worthy of the upgrades. Good luck whatever you decide.