Thinking about this from first principles, a banana is about:

• 75% water
• 20% carbohydrates (mostly sugar and a small amount of starch)
• Less than 5% combined fiber and protein
• Trace amount of fat (generally considered to be zero).

The anti-crystallization effect of guar gum works on the water component, so adding a small amount of guar gum actually should be of some benefit. The hard part is getting it all dispersed; you don't want to over-process the banana (you'd lose the creaminess) but you also need to hydrate the guar gum in order for it to work its magic. You also need to avoid over-thickening, which might be difficult even with the very small starch content in a typical banana.

Bananas convert more starch to sugar as they ripen, so I suspect you'd have more success trying this on very ripe bananas that are, essentially, little more than sugar and water to begin with.

Incorporating and maintaining air is something that I just don't think you're going to be able to do. Proteins and fats can both participate in foaming (egg whites being an example of the former, and heavy cream being the canonical example of the latter), but bananas have neither of those. There's simply nothing to whip, and thus nothing to stabilize, so neither xanthan gum nor any other emulsifier/stabilizer is going to do you much good here.

If you really want to incorporate air then you're going to need more than one ingredient. My instinct would be to say coconut milk, which has relatively high fat and pairs well with banana (and obviously is also vegan / non-dairy), but I can say from experience now that coconut milk just doesn't have enough fat. You could try cooking it down first to get the fat ratio up, or using coconut oil, but I'm not sure how it would taste.

Traditional ice cream is about 60% water, which means the amount of guar gum that most people seem to get the best results with (about 1 tsp/qt) is about 0.87% of the total water. For a small (100 g) banana, that's about 75 mL of water corresponding to very close to 1/8 tsp of guar gum.

So, I'd give this a try with 1/8 tsp of guar gum, forget the xanthan gum. Try adding it after you've frozen the banana for the first time, when you first pop it into the blender.

No guarantees, but, you can probably afford to waste one banana in the attempt.

Update:

I thought I'd give this a try myself since (a) it seemed pretty easy and (b) I can always use more non-dairy dessert recipes. Here's how it came out with 1/4 tsp of guar gum for 2 bananas, after being frozen for two days:

You can see that ice crystal formation is pretty minimal, and although some parts did harden a bit (much like ordinary homemade ice cream with low overrun), they returned to normal consistency within just a few minutes.

Of course it still tastes like frozen bananas, so if I were to do this again I think I'd probably try adding a few other flavours. Also note that even though the texture is decent, the colour is getting dark, and I have no doubt that this would eventually turn black after enough time in the freezer - and the lemon juice trick doesn't really work so well when the whole thing has been blended up.

Brine

Some assorted thoughts, before I try to guess an actual quantity:

Dependence on overrun shouldn't really be significant, as long as you're mixing well. The air doesn't add significant heat capacity, but it will make heat transfer less effective by providing some insulation, so you do need to mix well.

I can't imagine dependence on type of ice cream being that huge - whether you're using cream or egg yolks, you've still got plenty of water. I found a table for specific heats; it looks like eggs have a slightly lower heat capacity above freezing, and slightly higher after. I'm not sure about just the yolks, or about the latent heats of either. In any case, the properties of the ice cream custard will be somewhere between water and those. (Wildly generalizing, ice cream might be 1/5 sugar and 1/6 fat by weight.)

The more salt you have in your brine, the colder you can get it before it freezes. You won't be able to get it as cold if you don't have as much salt. I think you were slightly misinformed in chat: 0F (-17.7C) was defined (for whatever reason) using ammonium chloride. Saltwater brine (sodium chloride) can reach -21.1C (-6.0F) with 23.3% salt by weight (source: wikipedia). I believe the freezing point is roughly linear as a function of salt concentration; in any case, it'll be the most effective with the highest concentration.

You missed one factor that may well be bigger than anything you listed: heat coming in from the environment. This will depend on how well insulated your outside vessel is, the ambient temperature, and how much surface of the brine is exposed. (Also on how much you're gripping it with your hands, I suppose.)

As for quantities, I could try to ballpark something using specific heat/latent heat, but I think a far better method is to just look at the recipes for "ice cream in a bag" - the general idea of those is to put the ice cream mixture in a smaller plastic bag bag, inside a gallon bag full of icy brine, and shake. There are a lot of recipes using a sandwich or quart bag with 1/2 to 1 cup volume (probably ~120-240g) ice cream mixture, inside a gallon bag with at least 2-3 cups of crushed ice plus plenty of salt (maybe 300-600g? hard to say). I also found this one with the brine just in a bowl, using a pint (probably ~475g) of ice cream mixture and 3-4 pounds of ice. The latter is a little easier to estimate based on; it suggests for 100g of ice cream mixture you might need 300-350g of brine. (The other recipes are consistent with this.)

Mixing

This is really a separate question from all the brine stuff. If you want to get good ice cream, you absolutely must mix (churn). This both helps minimize crystal size and incorporate air. (Unlike whipping a liquid, mixing during freezing doesn't have to be as vigorous to get the air in.) It's possible that if you freeze it fast enough, you could make up for the air part with your pre-whipping, but I suspect you'll lose a bit of that volume during freezing. So at the very least, stirring will break up crystals, and it may well give you airier, fluffier ice cream too. (I'd be careful with that New York Times recipe I linked to; it's going to be prone to giving you unwanted crystallization if you don't take care with the "kneading", and may not leave a lot of overrun.)

General advice

I know you may have reasons not to do this, but in general, if you're willing to go to this much effort, you might as well buy an ice cream machine if you can. The part that you freeze will probably take up less freezer space than the ice/brine you'd otherwise use, and it'll also do the churning for you.