Are there any custom ROMs which let the user somehow filter Share menu lists? Based on " How can I customize the list of services that pops up when I choose to &quot;share&quot; something in an app? " answer, any app supporting ACTION_SEND intent will be listed in "Share" menu pop-up.

Are there any custom ROMs which let the user somehow filter those Share menu lists to exclude installed apps?
That's another XY problem question: you ask how to establish your suggested solution, instead to ask how to solve your real problem, which is your actual concern. For this task, you don't even need any Custom ROM.

Take a look at Andmade Share -- this app lets you customize your share menu on any ROM, not even root is needed:

 

As you can easily see, this app lets you...


Select (filter) the apps appearing in the list (as your question required)
Change the order in which they appear in the list (e.g. put your most frequently used apps on top)