I think it massively depends of what your T’s policy is on out of session and crisis contact.
With my T it would be a big massive no to email him, I would be breaking his boundaries and most likely would just be told to go to A&E and then next session I would imagine he would want to have a conversation about maintaining boundaries.
A lot of T’s however are open to being contacted if you’re in a bad place. It all depends on what their personal policy is, and just maybe prepare yourself that if you do email you may not get the response that you want.
If she is pretty flexible with booking additional sessions I would perhaps look to do that, and then discuss how you’re feeling in the session. I totally get the point about wanting to use your own coping skills, but when suicidal thoughts are becoming that intense, I think that is a perfectly acceptable reason to ask for help and see if there is an extra session available.