Sorry for the essay but here goes:
The swingweight will be affected, about 1 point lighter for every 5g in the grip. But you won't feel it much because it's between your hands; the club won't really feel head light. The grip is the fulcrum point in the club for the swing, while the swingweight scale places a fulcrum point on the shaft 19" or so, which doesn't have any meaning unless you're a hockey player. Because of leverage, the head weight is a lot more important than weight between the hands, just like applying force at the fulcrum point vs far from the fulcrum point of a lever will create different results.
If you're worried about total weight, they make lightweight grips (which I hate), as little as half the normal weight. It does not really have the same impact as adding weight to the head; using a light grip to make a club read D7 won't be nearly as head heavy as using lead tape. For shorter clubs like for women or juniors or ultralight overall clubs they work fine though.
For a better and more accurate method of matching clubs, look up MOI weighting; it's essentially what swingweight was designed to measure but without the simplification. However it's very involved and requires a lot of tinkering to set up a bag that way. Swingweighting works pretty well overall, and is the industry standard, it just isn't designed to factor in significant changes in grip and shaft weight, it's more about length and head weight. MOI weighting also brings into account the total weight, which can make 2 clubs at a D1 swing very differently.
Using tape might be a bit lighter than a dedicated midsize grip, but I don't see how it's good for the grip to stretch it, especially a cord grip. You can get a specific profile though, like making the bottom thicker to remove the taper. Other than that it's a hassle. Tape is not weightless itself, so it would still affect swingweight.
As for grip thickness affecting ball flight, it's a load of BS. I get the same shapes off midsize and standard. You might use a stronger or weaker grip on the club subconsciously or something, but it will not noticeably magnify hand action to change by one size. However, using an uncomfortable size will cause a lot of trouble. I have ML/L size hands, should be using standard grips based on fitting charts, but use midsize because it feels a lot better and doesn't require a tight grip to control.
Think about this: if most golfers slice the ball, and small grips encourage a draw, why wouldn't every driver sold today have ladies' grips on it? They use every imaginable method to encourage a draw, so the grip must not have an effect.
The average golfer can only barely feel a difference between 2 swingweight points. However I'm willing to bet he could tell the difference from a single wrap of tape in the grip. Make sure it's a comfortable size first and foremost, you'll hit it better.