Obviously it is your own decision, but they tend to be very good in taking care off stuff including "out of the norm" if your worry about that is the deciding factor pushing you toward taxi instead of other reasons.
Laptop etc. you could easily just take with you in a day bag. No biggie, right? Medication is one you will always want to keep "on" yourself, to minimize risk of abuse. No matter how low, the risk is not worth it for the non OTC heavier stuff. Medical supplies, aids etc? Nothing even close to new under the horizon for bell services, whether it being something very common or uncommon.
You don't even want to see the amount of medical stuff I have to schlap around.

But no biggie at all for smooth moving with a split stay. Including stuff like the large amount of tube feedings that can't be out in Florida weather without spoiling etc. If worried at all; simply talk to bell services the day before and if need be ask to speak to a manager to know what your options are. They've seen enough to find a way to make things work whenever possible.
With this switch you can easily hop on a DTD bus and walk to the check in desk from there, or walk from BWV to DHS and hop on a SSR bus from there.
Just made the exact same move myself a little over a week ago. Can't say anything other than perfect smooth move made and perfect care taken of my medical stuff, removing any and all risks. Did have some extra things to be put in place because of just getting out of hospital the night before the move but when looking at it from the normal-run-if-the-mill-move point of view can't say anything else but very smooth experience.