I think a shooting sling is a marvelous aid to field marksmanship from supported positions, but the 94 Winchester is probably not the best platform for it. The magazine tube is attached to the barrel by a clamp near the muzzle, so any pressure you put on the magazine tube by slinging up to a magazine tube mounted sling, will tend to deflect the barrel. What you need to do though is try it for yourself, and see if the difference is enough to cause you to miss with your unslug zero. If the benefit you get from slinging up outways the deflection you observe on the target, you're good to go. Either way, it will be clear to you in short order whether or not to pursue it.
You do have options of where to attach the front swivel though. If it was me, instead of using the common magazine band for the front sling swivel, I'd drill and tap the forend retaining band, and use a threaded sling swivel stud, Loctited in place. With the swivel attached to the forend band, the pressure that slinging up puts on the barrel is nearer the receiver, and should have less affect on the barrel.
So I just reread your post, and now I'm not so sure a shooting sling is what you're after. Your shots are made quickly, and probably from off-hand rather than from supported field positions, which is where the greatest benefit of slinging up is realized. Some folks claim to have a steadier hold off-hand with a shooting sling, but there's really no reason for that to be the case, whereas from a supported position the rifle is stabilized against a solid surface, such as your arm against your leg, or your your arm against the ground. Off hand is the most difficult position to shoot from, there's no getting around that, and the answer to the problem is practice. The one technique I've found to be beneficial is a little counter intuitive, since we're told to relax when we make a shot, but if you try it, you'll find that your sights freeze in place for a moment, and for a moment only, and you have to break the shot the instant you see it. So bring your rifle to bear, steady your sights on the target, then tense your stomach muscles hard. The sights freeze on the target, and you press, not snatch, the trigger, the shot breaks, and you score a hit. The most demanding part of this technique is good trigger control, because the temptation is to snatch the trigger. You have to press the trigger like its a warm summer day, you have all the time in the world, and not a care, but you have to do it in a compressed period of time, or the sights will once again be in motion, and you'll have to start over.
If you do intend to go ahead with the shooting sling, there are two I recommend, the first is the one I primarily use, the Safari Ching Sling from Galco International. The problem here is that Galco shipping charges to Canada are slightly outrageous, and you'll have to phone in your order. The second sling I recommend is Les Tam's 1907 military sling. This might very well be the best quality sling available anywhere in the world, and is custom made to your height and physique, yet its only $75 USD. Even if you choose to go with another brand, I strongly urge you to Google Les Tam's slings and read the information he has on his website. A military sling isn't quite as quick as the Safari Ching Sling, but its not that much slower once you get the hang of it, and you can sling up early, in preparation to shoot.
Hope this helps