Afraid I’m going to have to echo Beka. Sorry but if you want a litter trained rabbit you will one, need to give him constant access to the litter box; and two, plan for a different living arrangement. He is going to need more space. Is your bedroom rabbit proofed? If it is (it should be since he is obviously spending a lot of time loose in it) then once he is litter trained you might consider letting him have free range of your room. He’d still need something, a pen, or cage large enough for him and his litter box and dishes, space to lie down. This would give him a spot that is really his, but if he had free run of your room all day every day that could be on the smaller side. If he needs to spend time in his cage then it needs to meet that minimum requirement of space. Either way what you are describing is definitely too small.
In answer to your question though, yes I would pick up the soft bedding, it will only confuse him. Putting down a couple of big tiles is a quick and easy change which he may like, or if it is a solid plastic floor that would probably do for now anyway.
I suggest you take a look at the habitat section on this forum, lots of fabulous ideas there! Should give you somewhere to start planning, and show you how other people have worked around limited space, and/or limited funds (if you are dealing with either of those) and still designed and built living areas that work for there rabbits. Again if he can spend most of his time in your room the solution may be quite simple. Once you’ve had a chance to research what you need and checkout some of the great ideas you can always start a new thread asking any questions or for added suggestions to help with constructing your new home. (You’ll get more, and more relevant answers that way.)
I’m really happy he is being his happy playful self part of the day. Again, a little woory is always good, so much better to ask and find out something is nothing rather then not be concerned. I’m sure everyone looks forward to hearing how he progresses.
Since you are responsible and a little worry prone this is probably a good time to study up on what is going to happen when he starts going through his hormonal changes, … and begin to plan for his neutering. You’ve got a few months but expect you’ll be happier knowing what is happening as it happens. The House Rabbit Society’s site (I posted a link to it for you in another thread) and the Binky Bunny site are both terrific resources with reliable information you can count on. And of course, you can always ask on the forum.
I hope you enjoy each other Very Much! Best of Luck!