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A few months ago, I managed to catch a shiny abra, which I just brushed off and put in the PC because (as most of you probably know) catching a shiny with good IVs is phenomenally rare. However, after taking it to the IV NPC, I learned that it had a perfect 31 attack IV, which got me interested. I was thinking that, first of all, shinies are just flipping awesome and impressive, and could be a nice surprise as a physical sweeper (or maybe a mixed one), rather than a special sweeper.
It has a bashful nature (no affect on stats), has Inner Focus as its ability, and still has 0 EVs.

Anyway, enough about my case. I just need to know if an Alakazam could make a good sweeper
I just recently got into training/battling competitively, so I'm not entirely sure whether this is a good idea, what do you guys think? Thanks in advance.

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I love gimmick sets but this one might be a bit too ridiculous

I did some calcs for you, this should answer your question well

Alakazam at level 100, Adamant (+Attack -Special Attack)

31 IV's | 252 EV's in attack gives you a stat of 218
0 IV's | 0 EV's in Special attack gives you a stat of 247

Not even with it's attack boosted to the absolute maximum and it's special attack lowered to the lowest possible, could it's attack even come close to it's base special attack. And chances are your special attack IV isn't 0 so that's even more. of a difference.
But shiny's are cool, so if you insist on using it you should pull off a special set regardless of your attack IV

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What if I made him a mixed sweeper, focusing mainly on special attacks, but also using Drain Punch?  The move has good coverage as a fighting type attack, and restores HP.  Or should I go full special sweeper?
Alakazam should NEVER have any physical moves if you're playing seriously. It shouldn't even cross your mind.
252+ Atk Alakazam Drain Punch vs. 0 HP / 0 Def Aipom: 162-192 (64.5 - 76.4%) -- guaranteed 2HKO

I think it's a bad idea.