How can I best switch internally and uplevel?

You can explain on why you would want to move like lack of role progression, opportunities, tech stack etc. Explaining the switch more as move to help you grow and align with your interests rather than degrading your previous team. Normally companies don't Up level while switching team (I could be wrong here)

How does one approach switching teams internally without bad-mouthing the old team?

Just... don't bad-mouth them I guess? Do the classic "I really appreciate everything you've done for me, but it was time to find a new environment to discover more about myself." A lot of the insights in this discussion apply here as well: "How can I gracefully leave my job?"

And can we switch and up-level internally?

I'm very sure this is impossible, and I would be very worried about any org that lets this happen. If you're switching to a new team, by definition that team has less trust in you compared to your old team (unless something went really wrong). If your old team doesn't believe that you're operating at the next level yet, it's very unlikely that the new team would.

Since you're a senior engineer, promotions are going to start to be very lagging. Going from senior -> staff at a Big Tech company like Intuit is generally a 9-12 month lagging period: Staff engineers require a ton of scope within their teams, and it takes a lot of time to build up that trust and show your amazing capability.

And can we switch and up-level internally?

At Amazon and Google, this wasn't possible. It wasn't even possible to try to negotiate compensation even if the levels stayed the same (I attempted it when leaving Amazon).

Having said that, you can switch into a role with more responsibilities internally, even if your levels stay the same. If you perform well in these cases, you may be able to get promoted soon after switching teams.

I have heard of one other example where the destination manager promised to promote the person switching team within the next promo cycle. I'm hopeful for this person but IMO this feels very risky.