I think if I was you I'd be really careful about applying to LSE with computer science super-curriculars. Personally I was in a similar boat when I went to the LSE open day, as I wanted to study computer science at other unis but data science at LSE, and even with data science, there's hardly any computer science so that you might come across as not very well-suited to the course if you do apply with computer science supercurriculars in your PS.
Plus, if you're up against other candidates applying to Oxford who are solely talking about maths and computer science, the same thing is going to happen - economics may seem irrelevant and suggest you might not be the best applicant for the course. Even if Oxford doesn't value the PS as much as other things, it's not like they don't read it altogether. At the end of the day, all the unis you're applying to are super prestigious and they want to see that you care about THEIR subject, not about others.
Basically even the most computer science heavy course at LSE could be applied to without even talking about computer science and simply focusing on maths and stats. So honestly, I'd say there's little point in applying to LSE at all, especially if you want to maximise your chances of getting Oxford with 50% computer science. I suppose you could do something both economics and computer science related, like predicting stock prices with machine learning algorithms (obviously there's less basic ones than that), but the same problem of not being a fully suitable candidate both for LSE and Oxford still applies.
But good luck with whatever you do!