Do you think I should interview anyway?
Yes, go on the interview. It should not do any harm and you might get an excellent offer that makes you want to jump to the larger company. If they don't make an offer, or don't provide an offer good enough to make you want to leave, then stay where you are.
Moreover, is there a decent way of handling this interview as leverage to get a higher salary at my new job, without destroying my relationship with my current company?
An interview isn't really something to use as leverage IMO, as it isn't really anything tangible. If you try to use the interview as leverage with your current employer, you run some risks. The worst case scenario I can think of is that your current employer decides to terminate your employment (after all, you must be unhappy with the job if you're interviewing already, and they don't want unhappy employees) and you don't get the other job. If you get an offer from the other organization, you might be able to use that as leverage for a better salary from your current employer.
That said, after only a few months on the job, you probably haven't built up a lot of value with the organization you currently work for. Thus, it's more likely they will let you go to the new job than to try to keep you. However, if that gets you a raise and shorter commute, you are probably better off in the long run.
One caveat: Jumping ship after just a few months can be done once or twice in a career, but don't make a habit of it. While employers no longer expect you to stay at a job your whole working life, they don't want to hire people with track records of staying in a job for only a few months.