I think that perk trees connected to the traditional TES classes could solve this conflict seamlessly. These perk trees would require a prospective character to have expertise in more than one skill to qualify for and pick these perks. For example, Illusion and Speechcraft could be required to select certain Bard perks, Destruction and One-Handed could be used for some Spellsword perks, H2H (assuming its still in) and Acrobatics would be needed for Monk perks, Sneak and Alchemy for Assassin perks, etc. These are just examples and not all of them would require the same two skills, like a Assassin perk could be One-Handed Dagger and Sneak for a more stealthy assassination strike and Spellsword perks requiring Alteration and Block to allow a better, magic enhanced One-Handed block, all depending on what the function of the perk is. If a perk is usable in any combat style and not specified to a single style, like a combat perk that could work with One-Handed or Two-Handed, its requirement could simply be like Block + One-Handed/Two-Handed, so you could choose which style to use.
I think taking some inspiration from the TES classes for some perks would allow RPers to be a class of their choosing while allowing jacks-of-all-trades to dip into each class perk tree and take what they wish depending on their skill qualifications. I don't mean to presume Bethesda hasn't already solved this their own way, likely far superior to my musings, just wanted to do some brainstorming as to how this sort of thing could be implemented. If you have any comments, ideas on how this could be handled, some examples of Class specific perks, or whatever else, feel free to share.