The concept has been researched before. (Once you know the answer, google for it ...)
First there is old work by Book and Chandra, with the following abstract.
Summary. It is shown that for every finite-state automaton there
exists an equivalent nondeterministic automaton with a planar state
graph. However there exist finite-state automata with no equivalent
deterministic automaton with a planar state graph.
The example and argumentation given is exactly the one by Yuval in his answer!
Moreover they also consider the binary alphabet.
There is a 35-state inherently nonplanar deterministic automaton over
a 2-letter alphabet.
This work is continued rather recently by Bonfante and Deloup. They consider topological embeddings.
Informally the genus of a graph is the number of holes that have to be added to embed the graph a surface without crossing edges. Graphs with genus zero are planar. Then the genus of a language is the minimal genus of the automata for the language.
Theorem 9 (Genus-Based Hierarchy). There are regular languages of
arbitrarily large genus.
In the section "State-minimal automata versus genus-minimal automata" one finds the result, the proof of which is the first example given by Yuval (ten states to make the five state K5 language planar).
Proposition 7. There are deterministic automata with a genus strictly
lower than the genus of their corresponding minimal automaton.
The genus of regular languages, Mathematical Structures in Computer Science, 2018. doi 10.1017/S0960129516000037. Also ArXiv 1301.4981 (2013)
R.V. Book, A. K. Chandra, Inherently Nonplanar Automata, Acta informatica 6 (1976)