I asked for a raise after 2 years and my manager said my salary is "competitive for the market" , how do I push back?

So I've been at my current company just over two years. I came in at a decent salary but have taken on a lot more responsibility since then, basically doing the work of someone a level above me. I finally asked for a raise in my last one on one and my manager shut it down, saying my compensation is already competitive and budgets are tight right now.

The thing is, I've done some research on Glassdoor and Levels.fyi and my pay is actually on the lower end for my role and experience level. I have receipts.

My questions are: how do I go back to my manager with this data without it turning into an awkward standoff? Is it worth escalating to HR? And at what point do I just start looking externally and use an offer as leverage?

I genuinely like the team and don't want to leave if I don't have to, but staying quiet clearly hasn't worked. Has anyone been through this and actually gotten results without torching the relationship with their manager?

Any advice from people who have navigated this would be really appreciated.