How to install wood laminate flooring against existing hardwood transition in doorway?
You are almost certainly going to need to remove the existing transition strips and replace them with a different type. The upper portion of picture below shows what is most probably the configuration of existing strip. The lower portion shows the what you will need to transition to the new floor on the right. 



It was hard to tell from your picture if the hardwood floor was a full 3/4" type with the transition strip actually also 3/4" thick and joined to the hardwood by tongue and groove. If so then my picture is not going to be exactly correct. For now lets go with the existing hardwood being some type of engineered flooring that may be 3/8" or 7/16" thick like my picture.

In any case if you were able to remove the existing strips without damage AND if you had the proper tools you could probably cut the additional rabbet cut off the right side to accommodate the thickness of your laminate flooring. Short that possibility you will have to make or purchase the proper type of T type transition strip. With luck your engineered hardwood and the laminate flooring are the same thickness and then you will be in a much better position to purchase a more standard T strip that has the same size rabbet cut on both sides.

Removal and replacement of the existing transition strip looks like at a minimum that you will have to temporarily remove the door stop moldings as well. It the the new transition ends up being thinner (which is possible of you use one of the newer ones that are made of plastic) you may even want to fully replace the door stops so they can extend down to the new transition.

Note that the finishing touches that go with what seems like a relatively simple choice to rip out the old carpet can cascade to a lot of extra work and added expense. A new carpet may be an overall easier choice being as you are selling anyway and I think most people prefer carpet in a bedroom anyway.