Unfortunately, I made the critical mistake of assuming a 74161 is a 74161 is a 74161.

Turns out the Signetics variant does not operate in precisely the same manner as the linked VHDL description.
This problem succeeded in driving a colleague and I batshit crazy for a week. After factoring out signal integrity and test code as possible culprits, I dropped all assumptions and tested the chip's behavior manually. What I observed was that when ENP was transitioned HI-to-LO while CLK was LO, the counter would increment once on the following clock cycle before holding count.
I immediately thought, "...gotta be counterfeit lot. Surprise!" Turns out after finding the
Signetics-specific datasheet, the caveat was indirectly in the fine print:
(b) The HIGH-to-LOW transition of CEP or CET on the 74161 and 74160 should only occur while CP is HIGH for conventional operation.
Based on the quote above, the behavior I observed is technically undefined. TI's datasheet makes no mention of this whatsoever (although their timing diagram is consistent with the Signetics caveat).
Lesson learned. I wonder how many bugs were a result of this subtle nuance.