AASHTO/FHWA are completely unconcerned with Interstate business routes. Always have been.
Thus that's not a factor in any decision whether/not to add.
If it's only temporary signage during a construction project, then let's not add it; I'm with Jim here.
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In addition to Bickendan & Si's examples, I'll throw out that it's fairly common in Texas, when a toll road is involved, to have one number/designation on the mainline, and another number on the frontage roads. TX130/US183, GeoBushTpk/161, GeoBushTpk/190, TX183A/US183, and maybe more...