Creating a pragmatic, useable (not theory heavy) proxy for a currency basket for a set of economies
There is an economic literature on optimal currency areas. However, that is related to the adoption of a single currency.
As described in the question, a basket is just a new asset that is a weighted allocation to underlying currencies.
Once the weightings are fixed, the fair value of the basket is the weighted value of the component currencies, which can be updated in real time.
The only way that this basket changes anything in the real economy is that entities are willing to hold it and trade it.
The issue is straightforward: people and firms want their local currencies, not the basket. What they want is to translate the foreign currency to theirs.
Banks can offer quotes for any currency pair. It is only in the worst case that the quotes would have a bid/ask that is as wide as doing a pair of transactions versus a major currency (often USD, EUR). The only way a basket can improve matters is to have a tighter bid/offer versus the national currencies than they have versus the major currencies. How plausible is that outcome?
The weighting could be done in any number of ways, but the basket has to meet the constraint that people want to own it. The weighting has to reflect the ability to buy short-term financial assets in the underlying currencies. Realistically, that determination has to be done by the entities managing the basket.