Ok, the new screen is going main-branch (and default-on) in 1.005 (which will probably come out Monday). I've made a few additional improvements like with the instructions moving to the tooltips, etc, but to address the specifics raised here (leaving aside non-customization-related stuff like city map filters) :
Store/Inventory integration
Literally bringing in the buy interface would make this even more of a frankenstein's monster, but I did take ScrOBot's suggestion of making shift+right-click sell an unequipped or archived inventory item on the new interface.
Secondary filter dropdown
I _could_ add that, but there are two main reasons I think it unnecessary:
1) When you do two successive filters, it retains the order of the previous one in cases where the new one doesn't find a difference between two items. Of course, that doesn't help in cases where you'd like to do something like sort by +Regen but want +10 shields to "count" more than +1 Regen, but honestly I don't see how we can add that much additional info/control without making it even more super-crowded.
2) For loot generated in 1.005 or later, the variant-types (which use the same part name, but different graphics) will no longer have their own effects, effectively reducing the number of effects on most (say, 70%) parts by 1 and thus substantially reducing the need for this kind of advanced filtering.
Load/Save Loadouts
Again, could do it, but I'm concerned that the functionality will be so niche (the only really legit usage is to save, wipe clean, try something out, and then re-load it if you want to scrap the new approach) that a lot of players will look to it to do things it can't really do, like intelligently retain some kind of loadout concept across multiple missions despite some of the parts going to other slots, other exos, being sold, etc. And when they discover that it doesn't in fact do those things, they will ask why it's there in the first place