If I want to craft a weapon with a continuous 1st-level Frostbite, the item creation rules say it would be 2000 gp, the exact price of a +1 enhancement bonus. Would that count as a +1 enhancement bonus towards the +10 total, or would it be it's own thing?
In general, unless items are priced with a +X bonus, they are their own thing, and don't count as a +X enhancement.
2 things, however.
1. Frostbite is an instantaneous spell. You can't have a continuous version of it. Even if you could, the effect would apply to the wielder.
2. The item creation rules are explicit in that the GM gets final say in how custom items are priced
Many factors must be considered when determining the price of new magic items. The easiest way to come up with a price is to compare the new item to an item that is already priced, using that price as a guide. Otherwise, use the guidelines summarized on Table: Estimating Magic Item Gold Piece Values.
The correct way to price an item is by comparing its abilities to similar items (see Magic Item Gold Piece Values), and only if there are no similar items should you use the pricing formulas to determine an approximate price for the item. If you discover a loophole that allows an item to have an ability for a much lower price than is given for a comparable item, the GM should require using the price of the item, as that is the standard cost for such an effect. Most of these loopholes stem from trying to get unlimited uses per day of a spell effect from the "command word" or "use-activated or continuous" lines of Table: Estimating Magic Item Gold Piece Values.
What you are intending to do is create an item that a) does damage strictly greater (albeit non-lethal) than the +2 icy burst enhancement (assuming that the cl is >=2 and the damage multiplies on a crit, which touch spells normally do) and b) applies the fatigue condition with no save, and afaik fatigue infliction doesn't even appear on a weapon enhancement. Auto fatigue is extremely powerful - I would probably put that somewhere between +3 and +5 by itself. In any case, it is up to your GM how much this costs.