Hey!
As an international student, I can totally relate to the stress and the anxiety. I'm not sure that passing the SQE would boost your profile that much. Firms offer sponsorship for the SQE and it is something that they expect you to do before starting your TC. It would be more beneficial from a financial POV to get the sponsorship from firms instead of self-funding it. With self-funding, there is also the risk that you have a fancy degree but no place to work at. If I were you, I would focus on improving the applications and make them more targeted towards the firm. Explain what attracted you to the firm, go to events and speak to firm representatives to figure out what this firm has that others do not and why you would like to train at this firm rather than any other firm.
If you want more tips, I could suggest Commercial Law Academy's "writing successful applications" course, which provides guidance on how to talk about your experiences and the skills you have derived from them.