An adjective like 'political' when used with the noun 'stomach' always conveys the same idea. There is always some controversial decision to be made, and making it is likely to upset a large proportion if not all of a population and hence cause a backlash.
In this example the suggestion is that the populations of the key countries are not generally in favour of making the level of contributions required, and to do so is likely to have a 'political' cost at any subsequent election or referendum, and result in a vote against the government in power.
It is in fairly common usage.
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