I don't think not including BC would necessary.  I don't believe a BC PS4 would suffer from the same exploits as the PS3.  From what I understand, the primary issue with the PS3's keys is that the random numbers they'd use for the encryption weren't so random.  I'm sure they could fix such problems while still providing support for PS3 software.

And honestly, while BC is just a convenience for your consumer base, intended to spur hardware sales through included support for past libraries (as in, they'll choose PS4 because it supports PS3, etc., as opposed to choosing another console), I think it's paramount that they include it in their console.

I believe a lack of proper BC hurt them significantly this generation, though it was a move they had to make to lower costs (something that was hurting them even more).  With services like the PlayStation Network and Xbox Live essentially become platforms unto themselves, it'd be far harder to not provide support for such titles next generation and get away with it.

Next gen, I'm sure they'll at least support PS1/2 BC, given they'll still be using disc drives, and it'll merely be a software issue at that point, but I don't see why they wouldn't support PS3 titles as well.  It's been strongly hinted at that the PS4 will be Cell-based in order to facilitate PS3 BC, and I can't see this leak changing plans for that. 

The reusing random numbers is what caused the issue of them having the keys.  They can easily fix the random number issue in the current PS3 with little problems but that won't solve the actual problem-- they already have all of the PS3 (and apparently PSP) encryption keys. 

Fixing the random number issue is something they just should do but the damage is already done.  So I'm assuming this will be fixed in the PS4 but I'm just thinking having PS3 BC would essentially giving them an easy jailbreak type situation.  It wouldn't be the same success (for them) that the PS3 would be, but it would still be a success (for them).