The mod sets idea seems promising.
In the new mod section (not a sub menu from characters, please) you can create mod sets. Name them.
There you can select a character of your choice to put it on.
Or you go to characters, their mods. Click on sets and choose from. If mods are equipped somewhere, it doesn't matter. The whole set will be equipped for the current price.
The mods, you replace by equipping the set, just drop to your mod collection.
When browsing mods, a small icon on them shows, if they are part of a set or not.
Deleting sets doesn't delete the mods.
Searching for specific mods would also be nice.
Or suggestions, based on criteria.
For example, you've equipped three pieces on a character, and are now looking for the best triangle.
You can click on suggestions, set your filters (speed, offense,..) and it shows 3-5 results. No matter, if they are already equipped somewhere. If it is equipped, you could also visualize it. Either the characters icon, or just an indicator.