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Inquiries can hurt you in seeking new credit but again it all will depend.
But since you have been denied and BofA offered a secured. I would definitely go that route as it appears that you will need to work on your credit profile. If you recently applied for the BofA then they may not need to pull your credit again.
I would definitely go with the BofA secured, after 1 year they will graduate you (may not be exactly) but you can call the credit analyst and they can do that right on the spot unlike many other banks. When I got graduated I had to call and they did that right away, my deposit was back in my bank account 1 hour after I finished the call and my increased credit line was also immediately available, the new card arrived 10 days later.
At this point, time is the only thing that can help you.