As far as the rules are concerned, no. You've already made the distinction - Beguiling Defenses only works against being Charmed, and Suggestion doesn't Charm. You are immune to Suggestion, since it specifies that

Creatures that can't be charmed are immune to this effect.

You can, of course, talk to your DM and ask them to rule that Beguiling Defenses works fully against Suggestion. There's definitely a case for it, considering the flavour:

your patron teaches you how to turn the mind-affecting magic of your enemies against them.

And the fact that Beguiling Defenses makes you immune to Suggestion, which suggests (pun intended) that it should work, even though it technically doesn't.

You’re misreading dispel magic. It can be used to dispel, or it can be used to counterspell, and these are two separate things.

Dispelling ends an effect already in place. It occurs after a spell has completed casting. By definition, an instantaneous effect ends the instant casting is complete; as such, there is nothing left to get dispelled in the first place: there is no effect to end.

Counterspelling prevents a spell from being cast; the effect never happens. You cannot counterspell a spell already cast, regardless of its duration. Instead, you interrupt the casting so that the casting never completes, and the effect never happens. The line about what kinds of effects you can and cannot dispel doesn’t apply, because counterspelling doesn’t do anything to effects, and isn’t dispelling.

For the record, counterspelling, whether with dispel magic or by using the appropriate spell to counter whatever they’re trying to cast, is not the only way to disrupt spellcasting. You can also deal damage to them, either with a continuous-damage effect or with a readied or immediate action to hit them while they’re casting. Regardless of approach, though, the idea is to prevent the casting from completing, so that the effect never comes into play.

One of the most effective approaches is known as counternuking, and involves readying, as if to counterspell, but instead of readying dispel magic, you just ready the most reliable, high-damage nuke you can. When they try to cast, you blast them, and force them to make an unreasonably-difficult Concentration check, which they’ll almost certainly fail. Then you get to hit them and stop their spell. This tactic is effective, but may lead to a more boring game as spellcasters spend all their time waiting for others to move and are too afraid to ever cast their spells.