Ah-hah!
I didn't know the native browser was the only one that pushes pages to e-ink. It makes sense when you think about it, but I hadn't!
I installed RoboForm, which apparently acts just the same as Last Pass, giving me the option at every turn to continue with the (native) browser or RoboForm. So far the RF pages are incorrectly formatted, but that might be because I haven't paid for it yet.
Because I am loving the send-to-e-ink feature, I will probably stick with what EnTourage gave me, although I will certainly use one of the password managers as well, so I'll continue with the extra clicks.