Oh dear lawd, I managed to beat this High Score Challenge purely by accident. I was lazy and didn't want to spend any jewels so I just repeated the challenge to get a few bars of special gauge (3+ from the chest and then one or two more from the boss). On my last round of filling it up, I noticed I got it almost filled and so I thought "might as well start attacking", but at the end of my first round (when the boss still had, like +2 on its health bar) I noticed my Friend medal wasn't the Axel medal that is shared in my party.
Anyhoo, as I didn't have Axel, at first I just thought I wouldn't bother spending any more special gauge since I'd want to try to challenge with Axel. So I didn't use any special attacks on my second turn. Then the Friend Medal BC Mickey's Sleep Activated and the boss couldn't attack after the second turn and I was like... hmm... maybe I should try this, lol. I didn't think I had any change, since I was somewhat low on health and had a tier 3 def debuff on me + the boss still had, like, +1 on its health bar. I somewhat carefully used Piglet, Illustr. Cid (with a 44% guilt) and Lexaeus and got the enemy down to 2.9 health bars or something.
And then, after the third turn, Def Boost II activated twice when the card boss did the double attack, and I didn't die (if even one of them had hit at full strength, I'd be dead). So begins the fourth turn. I used Piglet again for the buff and just normal attack with whatever medal I had on the second & third spot (Belle & Demyx for the 7%, I think?) At this point it still had, like, 2.5 bars of health left, so I wasn't too enthusiastic about my chances to win considering I only had 3 medals left. I used Cid as my fourth medal, its Attack Boost II activated and blasted away ~1.8-2 health bars. The boss was left with ~a little over half its health bar, I had two medals left (Lexaues and the Friend Medal). Lexaeus was pretty weak against it, but BC Mickey helped take away the few inches of health bar it still had left.
Too bad I didn't pick the Axel for the extra 7% because I wasn't meaning to actually beat it on this try. Though, the only reason why I did was because of the Sleep on the BC Mickey that I happened to have chosen as my Friend medal. So I guess it all worked out perfectly, hehe.
EDIT: I broke 40k pretty easily with Piglet + Illustrated Cid (or one of the Illustrated avatar board characters) with Attack Boost II & 44% guilt.
EDIT#2: Also had Poison+ on Belle activate a couple of times. Without those, I would have been screwed.