Hi Ben - Welcome to TF! I'm with Cthulhu. Very, very carefully repot it. Not because of the soil right now. As the late, great Phil Mann used to say "Tuberous Drosera would grow in a wet sock if you let them." The reason I'm suggesting repotting is that transitioning your plant into dormancy without rotting the new tuber will be much more challenging in a highly water-holding soil. It can be done, to be sure, but it sounds like you are a little new to this group of plants so there is no reason to add an extra challenge at that stage.
You'll need to follow the stolon down to what will likely be a shriveling tuber from last summer. It is this main shoot that you do not want to snap. The lateral roots that come off the sides of this stolon are less critical. If you find the tuber is soft, or looks to be dying, that's okay. They replace their tubers every year, and it will have translocated much of its stored energy to the shoots at this time of year. Depending on how far your plant has progressed in its seasonal growth, you may find a second thick stolon descending to produce next summer's tuber. It would be best not to snap that if you find one, but it's pretty early in season for that. Mostly, I wanted to give you a heads up.
Remember, you won't need bare root it. Even if you leave a core of the peaty soil around all of the stolon and roots, that will be fine. It will still allow the plant to dry more appropriately at the end of the season. Good luck! And if you have any other questions, please just ask. I love this group of plants, and I'm always willing to help out folks who are trying to grow them.