After talking to another classicist, I can offer some thoughts, though sadly without definitive sources.

It seems unlikely that the two are related, for various reasons:

• Contracted perfects are extremely rare with the syncopated ending: amārunt and amāvere are both possible for the third person plural perfect, but *amāre generally isn't. Historically, the contracted perfect originated from a sound change ivi > ī, and was extended by analogy to other forms; since -erunt lacks an i to trigger this change, it was never the most popular form to contract (all instances were by analogy).
• Historical infinitives never seem to alternate with perfect forms: if one phrase uses a historical infinitive, the phrases around it generally will, too, regardless of person and number. You'll almost never see a single historical infinitive on its own within a larger text.
• Historical infinitives are restricted to certain contexts, and are mainly only used by historians (hence the name), not by other authors. Perfect forms on the other hand appear everywhere.
• Historical infinitives generally denote repeated, habitual, or ongoing action—in other words, they have imperfect semantics, not perfect.

After talking to another classicist, I can offer some thoughts, though sadly without definitive sources.

It seems unlikely that the two are related, for various reasons:

• Contracted perfects are extremely rare with the syncopated ending: amārunt and amāvere are both possible for the third person plural perfect, but *amāre generally isn't. Historically, the contracted perfect originated from a sound change ivi > ī, and was extended by analogy to other forms; since -erunt lacks an i to trigger this change, it was never the most popular form to contract (all instances were by analogy).
• Historical infinitives never seem to alternate with perfect forms: if one phrase uses a historical infinitive, the phrases around it generally will, too, regardless of person and number. You'll (almost?) never see a single historical infinitive on its own within a larger text.
• Historical infinitives are restricted to certain contexts, and are mainly only used by historians (hence the name), not by other authors. Perfect forms on the other hand appear everywhere.
• Historical infinitives generally denote repeated, habitual, or ongoing action—in other words, they have imperfect semantics, not perfect.

All in all, it seems to be a coincidence. Perhaps a very unlikely one, but a coincidence nevertheless.