It seems that BIBM just might possibly be (at least one of) the culprits here, shocking though that sounds -- it's a truly excellent product, after all!
Here's why I think that possibility exists: I finally managed to fix the problem using BoyAns.net Visual BCD Editor (stay away from EasyBCD!). The problem turned out to be that something kept altering the target partition from volume # 2 (Windows 7 Pro) to volume # 4 (Windows 10 Pro and it's awful bootloader). It seems very likely that it was Windows 10 itself doing that, but I'm not sure.
Anyway, my changes bypassed BIBM (as expected, since the boot partition was set to volume 2, while BIBM resides on volume 1), so once I verified that the problem was fixed by booting both OS's successfully, I re-activated BIBM v1.35c. But when I rebooted and selected Win 7 from it's boot menu, all my problems were suddenly back again! It would NOT boot Win 7, and instead it immediately tried to boot Win 10! And again, the Win 10 boot loader failed to boot Win 7 (the computer restarted instead). And when I looked at the Win 7 BCD again with the Visual BCD Editor, sure enough I found that the Win 7 boot entry pointed to volume 4 again!
Now, I'm well aware that that's far from proof that BIBM is to blame, but there certainly seems to be something highly undesirable going on. It's probably still Win 10's fault, but since I can only get Win 7 & Win 10 to dual-boot by disabling BIBM and using the Win 7 boot loader, perhaps BIBM is somehow involved after all?