You can insert a \t switch into the cross-reference's REF field to remove the "Appendix " part of the label.
(Sorry, I had forgotten about that). I have left the other two approaches below just in case they are applicable to some other variation on this problem.
One way, although you still can't use the cross-reference selection tool, and you will typically lose other Word crossref/tables/index features as well, is to do this:
Use
Appendix { SEQ App \*Alphabetic }"
to insert each Appendix label, as you have already tried.
Bookmark each Appendix label that you need to reference (it is only necessary that you bookmark the { SEQ } field).
So e.g. bookmark Appendix labels A-G as Violet, Indigo, Blue, Green, Yellow, Orange, Red.
Define your reference as
Appendixes { SEQ App Indigo \*Alphabetic }-{ SEQ App Red \*Alphabetic }
I am not convinced that it is actually any more maintainable to do this than to insert the cross-reference completely manually and maintain it manually. For example, if you reduce the range (so it's B-F), then you either have to re-bookmark F as "Red" or you have to change the second SEQ to { SEQ App Orange *Alphabetic }. This is a general problem with cross-referencing but it seems to me to get worse when you move away from the familiarity of the built-in mechanisms.
There is another approach that might just about work as long as everyone working on the document knows what is going on. It assumes LISTNUM is reliable (it seems to be a bit quirky):
Use multilist numbering to apply the full "Appendix A" label. Assume that you use Level 1 for that number.
Set up Level 9 (say) so that it consists of the number from Level 1 (using the "Include level number from" dropdown) and nothing else. Format the "number" as A,B,C...
At the beginning of each Appendix that you need to cross-reference, insert the following field
{ LISTNUM \l 9 }
This should display the same letter as is in the Appendix label.
Select the LISTNUM field, and insert a bookmark - e.g. use Violet...Red for A-G as before.
Format the Field/result as Hidden
Define your reference as
Appendixes { Indigo \*Charformat }-{ Red \*Charformat }
Now we're back to using Ref fields you may also be able to use other things such as \h.
Ideally, it would be possible to set the bookmark and hide the result of the LISTNUM using
{ SET Indigo "{ LISTNUM \l 9 }" }
but the { LISTNUM } result appears to be blank whenever it is nested inside another field.
The *Charformat is there to "counteract" the Hidden formatting.