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US citizen here. My answer to my own question leads towards "no". No harm in asking.
I have means to obtain a residence permit in Spain.
Can I then overstay the 90/180-day visa-free stay my US passport enables in Bulgaria?
Assume for answering the question that Bulgaria never becomes part of Schengen.
A third-country national who does not have a Bulgarian residence permit or Bulgarian type D long-stay visa is limited to 90 days presence in Bulgaria in any 180-day period. Nationals of some countries must have a type-C short-stay visa. The US is one of several countries whose citizens are not subject to this requirement.
Someone who normally would need a visa for a short stay can substitute (for example) a visa from a Schengen state, Romania, or Cyprus, or a residence permit. The foreign document serves as an exemption from the visa requirement, not as an authorization of stay. Thus, for a US citizen, the foreign document has no effect: there is no visa requirement to in any event.
Therefore, the answer to your question is indeed no. A US citizen's stay in Bulgaria is treated the same no matter what.
