Good morning RicTrim -
Cutting around the outer edge of the carpet below the bed could be an issue as you mentioned. The three sides would not be anchored down. The result could be the bed balling up the carpet underneath and jamming the slide.
The first thing I would do is talk to a carpet installer and have them look at the job. Get their opinion on securing the new flooring to the deck of your Monty. These are small jobs for the pros, if the area in which they are working is clear of furniture and clean.
I am going to remodel the bedroom in our 2008 Monty, and I have looked at this project which will require the complete removal of the closet and bed. These are my observations and how I will approach my job!
As mentioned by Mikendebbie, there are two rollers under the bed frame at the foot of the bed that cause the bed to roll across the carpeted/vinyl flooring. Check underneath the frame and you should see them there.
Once the slide is out, and if there have been no modifications, you can remove the entire bed. The mattress is easy, but there are other components to manage. You will have the deck the mattress rests on, and the mechanisms used to move the slide. I have a hydraulic slide, so that component, the ram, must be removed.
The bed's headboard is held to the slide with a few screws unless there had been a modification and additional fasteners were added. You might be able to remove the bedframe without removing the headboard. Disassembled the bed to the greatest extent possible and then moved it out of the room. You will then have complete access to your floor.
Now it is time to remove the carpet. Here is a good video discussing the removal of carpet from an RV. Not much different than removing carpet from a stick-built home. You may find that portions of the carpet are under interior side panels. The carpet should not be under the wall frame. Also, there could be a zillion staples to be removed.
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For the replacement of the carpet, I will go to the pros! I would suggest having them look at your project prior and hear how they would approach the job.
If you do the majority of the dismantling and replacement of the bed. The costs to have the carpet should be very low; and putting the bedroom back in order a fun project!
When reinstalling the bed, I really like the use of the rollers. However, I have seen DIY projects where plastic runners/skis were used. My lovely bride would like modifications under the bed to enhance storage and easy access.
Anyhow, this is my approach and what assessments I made. Best of luck and I would like to see photos of your project, if you don't mind sharing!
Cheers