He noted that he was looking to do a dungeon crawl, where the party was a character. (emphasis mine)
This sent my brain in a spin: USR seems perfect for this, although I have yet to try these ideas out.
Option 1: Standard USR Character
Your Action die represents your fighter, your Wits die represents your magic-user and your Ego die represents your Cleric.
Give each stat a specialism. Probably something like Melee (A+2), Arcane spells (W+2), Divine Spells (Ego + 2).
Calculate HP for the party as if it were a single character (Action + Wits, so yes, your M-U will bring down the party average, as they should) and assume that damage is equally distributed.
Combat would be handled as normal but would be very abstract. It also means no one character will die, and you'll end in a TPK when reduced to 0.
Option 2: Standard USR Character version 2
As above, but roll HP for each stat (member of the party). All participate in combat individually. Damage goes to the individuals. At 0, that member of the party is dead.
This could require fiddling with how you handle NPCs and monsters to make combat fair for both sides.
Option 3: Replace the Standard Stats with Classes
If you stick with the standard character creation, you're stuck with a max of 3 members of the party. Now, even if you're not fond of the thief, it still leaves out illusionists, bards, monks, etc.
Rename the stats, Fighter, Magic User, Cleric, or what have you, and add as many more as you'd like.
Assign a die to each. You probably don't want to go much above d12 for a single stat (and I would limit that to just 1 of the members), unless you want a truly superhero group or below a d6 for the party's classed members. Obviously, do what you want. d30 for your fighter? Sure, maybe he's Conan, or He-Man or something.
Give each a specialism, something class appropriate.
In this case, for HP, I would go with a lump - roll all the dice and add together. This really allows the party to be played as a single character. It also gives them staying power - maybe too much if you like life/death of your characters to hang in balance - you might drop the highest and lowest or something, to make a weaker or stronger party as desired.
As my notes in Option 2 imply, I'm not fond of the idea of tweaking the rules for individual HP per stat. The abstract combat really appeals to me.
As Ovy noted in our discussion, you could assign A: d20 to a group of orcs. That "group" is just 1 combat die roll, as is the party's combat roll. Just like standard USR.
I love the simplicity in that; let your narrative describe the details of who does what, not the mechanics.