I sympathise with you - I'm in a similar situation, where there is not a tradition of MBAs in my firm, and where an MBA would be viewed as a one-way ticket to leave the firm.
Fortunately in my role I change line managers every year or two, which means that whilst my current line manager is a "no go" for recommendations, I can get two or three old line managers with whom I have a good relationship to provide feedback.
The fact that your line manager went to HR to check if a LoR was OK worries me a little; my company has a similar policy, but based on the strength of the relationship with the recommenders that I asked, they wrote one anyway as a favour, without getting HR involved.
Have you considered going for 'historical' line managers? How about people involved in any extra-curricular / volunteering activities that you do?
I would also suggest that you might be being unrealistic with your number of applications. If you KNOW that recommendations will be a problem, is applying to six schools not a bit optimistic?
I know that if I had asked my recommenders to write six letters of recommendation they'd have told me to get stuffed. Maybe try considering applying to fewer schools? Yes, it's not fair that others get to apply to six whilst you don't, but I'd suggest that it's your best shot of at least getting some applications in. I've only applied to three places, and it's exactly because to apply to any more would be stretching the goodwill of the recommenders that I have.
Finally... as has been suggested by others, you could help by drafting a letter of recommendation / provide bullet point suggestions, based on:
- Written performance feedback that you've already received from these people
- Letters of recommendation that they've already produced for other schools
Don't be tempted to write the letters yourself, aside from it being ethically questionable, people have very different writing styles so it becomes very obvious to the adcom if you have written it yourself. Copying and pasting content from the above two sources keeps you honest, keeps the writing style different, and provides a good frame from which the recommender can complete the recommendation.