I wouldn't call bleeding after vag exam 'normal' unless the exam was rough, or a sweep was done. Both of which are highly likely for a woman getting a needless VE at 37wks. Of course, the bleeding you describe may not be 'normal' IMO, but seems to be non-problematic anyway.
Definitely agree that informed consent is needed here. Maybe this mom would be happy to have VE, even knowing they are needless and do not provide any important information. But the choice should be hers, something she wants--NOT something done to her routinely and without consent.