I see people talk about self-funding the SQE after unsuccessful rounds, but does the SQE actually make someone a more attractive candidate for these types of firms? You've mentioned
Cooley, Simmons, and
Hogan Lovells, but at these sorts of places I'd wager that the saved cost of not having to sponsor you through their exams (if you self-fund the qualifications) is unlikely to move the needle between you and another candidate. Even if it demonstrates resilience/perseverance/etc, I don't think it adds anything to an application that is unique to the SQE and can't be shown through other means (that you don't have to pay for)
With solid academics (as you say), maybe your time would be better spent working on the lack of work experience that you've identified as your issue - whatever it is that is stopping you from advancing through initial application stages, I don't see how the SQE is going to meaningfully change your profile/help your application? Obviously I get that it's frustrating to feel like you've hit a brick wall/that you're not making any progress but if you feel that you're lacking quality experience surely it'd be a better use of your time to try and address this first before rushing into another qualification especially if finances are an issue?