Tips for color correcting footage shot inside with bright windows always in shot?

Mike's got good comments, especially neither over-thinking, or going for perfection Wise statements.

I work for/with/teach pro colorists, and quite often they need to match a set of cameras where there is a big image quality diffference between them. You cannot make the "poor" footage as good as the really hot stuff ... so you do as best as possible in the time allowed for the job! ... then bring the 'good' stuff down close enough the clip-to-clip progress down the sequence doen't have any visual 'jumps' to it.

So rest assured, even in "big budget" productions, you can be given less than perfect media.

Typically, broadcast or cinema companies doing this would have put large sheets of neutral density material over the windows, along with boosting interior lighting, so that you don't get that blown out view through the windows. Small operations, and especially one-person run & gun, may not have time/budget/materials for such things.

So blown out windows is going to happen. Learning to do what you can while shooting, and what you can in post, is important. And what the limitations are that you can't get around.

As to how far you can take it ... what's the media? Format/codec, produced by what, and especially, what is the bit depth ... 8 or 10? 10 bit media can be pushed a lot farther before it breaks up, especially if it's intraframe ... something like ProRes. H.264/5 "long-gop" 8 bit is especially prone to displaying artifacting like banding or macro blocks on being pushed much.

So Mike's suggestion to go with Temp/Tint first is good. Then do tonal work, but you really need to have scopes up for all of this. Vectorscope for color saturation and vector (hue angle around the color wheel) and Parade RGB and Waveform (YC no Chroma, for "pure" tonal information) are what I normally have up.

In say Parade RGB, or the Waveform, if you see a solid line across part of the top of the image, that would indicate clipping ... there's a hard limit to data and nothing above that line. Pulling the whites down lower won't get any data back.

So set the blacks to at or just above zero, and whites at the point where you lose data.

Then between there are different tools that all behave a bit differently. Basic tab's Shadow and Highlight controls actually move everything but the end points, they just move their named section more than the rest. So often, you move them against each other to get a more targeted area of change.

Same with the color wheels ... "shadows" moves the left side (dark) area of the scopes and image more, end point included, "highlights" works on the top of the media ... but both have some affect across the image. And the 'gamma' or midtone control touches everything but the end points. So you typically say push the shadow slider up while moving the mids slider down to lighten shadows without brightening midtones. 

Or move Shadows color to the blue/cyan side to cool things down visually, while pushing the Mids wheel the opposite direct a touch so your mids/skin tones don't go cool.

The Creative tab's Tint controls are really handy ... especially if you want to correct a shadow color cast, for instance. Slide the control slider between them to the left, so it the shadow tint control is more limited to the shadows, and then use the shadow tint control to fix your shadow color cast.

For your image, I'd do temp/tint to get highights color acceptable (not perfect) ... then do tonal work for the black/white point, then try to get the better 'feel' of the image in shadow, mid, and highlights placement. There are some things you can do with blown out areas, but that takes a bit of experience and a lot of patience especially beginning to try fixing it.

But only goes a little ways in reality.

Temperature and Tint should do most of the heavy lifting to reduce the overly warm interior. Then if the outside becomes very cool, I would hue shift the highlights using the Color Wheels but if that is to broad an adjustment I would use a HSL Secondary adjustment and play with the settings using a fairly heavy blur. Make sure to check the box next to the 'White/Black' dropdown so you can easyily see what will be affected.

Don't overthink it - there is no such thing as a perfect image. Toggle off your correction, if its better than before then you've made an improvement - don't chase perfection.