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Are there any works where authors tried to define weighted finite state automata over hyperreals (or a similar object allowing for infinite and infinitesimal values) in an attempt to make automata properly behave for any assignment of weights?

In the probability semiring, for instance, epsilon loops with weight $\geq 1$ present a problem since they define an automaton with infinite values (as explained in this paper). An introduction of hyperreals can possibly make such an automaton properly defined, and, furthermore, normalizable (meaning that there exists such a quantity $Z$ that sum values of the automaton on all sequences divided by $Z$ equals to $1$).

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