Does Saving Barrier allow you to stay in Ascension?

I don't believe so, no.

However, I have experienced and found funny instances of non Ascended Ascension via other ways.

If you're after a way to keep Ascension going, you need to look at more exotic ways of dying and activating Ascension. That's all I'll say over the matter.

Yes, If your health is below the trigger threshold you will get your barriers back from saving barrier before losing your ascended state. This only works if you enter ascended state with low health and already used up saving barrier, and your health will naturally regenerate before you can use it more than twice at absolute maximum, with one use being immediate, and the next between the 15 second recharge and the point your health reaches over the trigger threshold.

Yes, but there is no way to sustain critical low health to use it more than once reliably.

Well, it's not the most easiest way to execute it, because of lot of it is down to very random chances involved. But... If you want a full round of uninterrupted Ascension (zero cooldown), you can attempt the following (only experienced on 1.09, dunno if still exists in 1.10 yet):

1. Obtain Ascension readiness.
2. Get taken down to a sliver of health.
3. Have an Adhi (or something else that grabs you) nearby.
4. Activate Ascension just as they grab you (this it the crazy chance bit as most of the time Ascension will trigger first and prevent the grab).
5. The grab will kill you despite the Ascension activation. (It's only invulnerability to weapons and explosive fire, not grabs)
6. On being revived (whether your own or by someone else), you will have perma Ascension that round. You can lose shields and Ascension cooldown (maybe DR too?) won't go. (Can't remember if Shields were utilised or not either, as again, it doesn't happen often due to crazy chances involved to make it happen) But during that round, you have zero cooldown without worry of shield collapse.

Yes, as said, it's exotic ways to die. And no, not something you can easily repeat. But if you get it, you'll soon notice being unable to recharge to get into Ascension and all cooldowns are always off. ~600 hours of Kineticist gives you such experiences.

Oh, having Saving Barrier or the other evolution won't matter to this "bug". But I personally recommend Saving Barrier in the end.

Well if you're gonna be *logical* about things, sheesh.