Quickie: best weapons for an Adept in ME3?

It does not really matter. Depending on your preferred Adept build and squadmates, you are pretty much playing without shooting a weapon anyway.

Basic Adept builds will be using Throw and Warp, and either Singularity or Pull (I prefer Pull), but mostly Throw and Warp. If you use Liara (Singularity) and Javik (Pull) for most of the game, you will rarely use your own Singularity/Pull. You end up just spamming Throw on health-only enemies, and Warp+Throw (and maybe Cluster Grenades) on everything else.

That being said, Pistols are generally better than SMGs:

• Scorpion (Sur'Kesh) can be good for its stagger when it hits.
• Executioner (DLC) because it is essentially a Mantis/Widow sniper rifle in handgun form.
• Acolyte Pistol (DLC). It is a charged-shot weapon, but it has a crazy 500% shield/barrier multiplier (meaning it deals 5x damage to shields and barriers).
• The "real" Suppressor (Citadel DLC) is always good if you can score headshots. (In the Ambush section of the Citadel DLC, you get a "fake" Suppressor that is nothing like the "real" Suppressor. The "fake" Suppressor is a noticeably a lot heavier, and I think it deals less damage.)

You better off having a light assault rifle for it. A cerberus harrier is OP so that would be good. A Saber is great (even just for the sound it makes) but you only get that later on in the game. Honestly for SMG's I thought the hornet was pretty good as well. Maybe the N7 Valkry.

Be happy to share my preferences for an Adept weapon loadout and strategy.
You are correct that most of your focus should be on keeping the cooldown bonus optimal. However, it doesn't necessarily need to be 200% exactly due to the fact that recharge bonus' have diminishing returns. If you wanted to enter missions with, say a 150% cooldown bonus from your loadout - you could and would still be effective as a caster and detonator.

With that in mind, if you wanted to carry an assault rifle, you could use the Mattock. My Adept had it for most of the run. It's a great all around gun because it is accurate, usable at any range, easy to fire from the hip, and great against armor.

That being said, pistols are my favorite weapon type in the game. I wish I could bring 2 into battle. As a biotic, you really want a pistol in your hand because of the synergy with the power magnifier mod. My two fav pistols are the Talon (150% bonus vs barrier/armor) and the Scorpion. The former because as a classical Adept (I assume you're not going with an unconventional build), you will want to maintain your distance on the battlefield to optimize crowd control. The Talon is handy in case you're forced into tight quarters (Omega, Earth) or get flanked (Geth Hunter, Husks). Fantastic stopping power especially when paired with squad ammo (James, Ashley).

As for the Scorpion, it's the best pistol for bosses IMO. The gun is accurate, can stagger and ammo power success rate is pretty much 100% on hit. I use the Scorpion against the bosses that rush you, such as Geth Primes, Brutes and Banshees. When they get close, or when a Banshee is charged, you'll want to unload a few rounds, turn around, run and find cover to keep distance at a maximum (again assuming this is a classical Adept build).

If you ever feel like mixing it up with a sniper rifle, shotgun or assault rifle just for kicks - you can on the N7 side missions, which are a shorter and easier missions. I may be an Adept, but sometimes I just want to shoot the Saber at Cerberus.

A high powered but light weight pistol.

git gud.