I would have think that Catherine would be fine with Bash if things had remained the same as early Season 2, because as you said, the relationship did seem to get better, but given how things were left by the end of the season and the animosity there, I'm not sure Catherine now would be okay with Bash still in the castle if Francis were to die. Unless Catherine has some reconciliation with Mary and Francis.
Even if Bash doesn't want anything, with Francis gone, so is his position and protection, and unless Catherine is back on their side again, I don't think she would want Bash around, people vying for control will see Bash as a threat, and Catherine being strategic as she is, would move to get him out of the way. She may not kill him, but I have a feeling she'd still lock him up somewhere.
btw, can I be added to the list?
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"I think I found my place on the team,
I'm the BLOCKER"
JEFFREY MACE|PATRIOT
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