MtheGM:
we are talking of a period of only a few hours at most, not any extended period.
No, we are not talking "a few hours at the most" the original question (as you posted it) states:
Each has a currency called the poly, worth 100 cents of its own currency; but by agreement a Polythene poly is worth 95 cents in Polydor, & vice versa. which suggests that they are established and ongoing exchange rates.
As shown above, this could not continue without resulting in ruin for the countries/banks etc. As such, it would never happen.
If there is a small difference in rates (such as when the Irish punt was remove from parity with the English pound) then you will find businesses on one side of the border willing to accept notes from the other side of the border, but giving change in the local currency. Businesses on the other side of the border will not reciprocate (unless their mark-ups) are sufficient to ensure they can still make a profit even taking into account their loss on the exchange.

In order for the transaction to work as set, each innkeeper would need to keep a stock of the 'lower-valued' notes to give as change. Where is he going to get these if the customers insist on using the 'higher value' notes?

Suppose the two innkeepers know one another, and decide to meet at the border to exchange unwanted currency after visit from this miraculous drinker. Neither is going to accept that they must take a loss on the deal by giving 105 to get 100. Even if they agree to exchange the notes at parity (100 for 100) they will quickly realise (as readers of "weekend" failed to) that this situation could not continue as it was costing them money.

A similar thing happened in UK (at least twice) when we went off the 'gold standard' people quickly retained gold sovereigns, while spending the paper money which was seen to have lower value. Similarly, in the 1970s it was realised that UK silver coinage was of more value (for its metal content) than the face value of the coins. A local Chines gift shop was buying any pre-1947 silver coins at 8 times face value. Staff in banks (and here I speak from first hand knowledge) quickly realised that it took only a few minutes to sort through £100 in 'silver' and extract any pre-'47 coins.These were replaced by an equal value in notes from ones pocket (often checked by a colleague), and the coins then 'cashed in' at the Chinese gift shop, to see a tidy profit. It almost became a competition to see who would serve those businesses known to bank a lot of coins.

See Gresham's law