Ok so right now I am going through the Jody Fisher Jazz series, so I am using this as the basis for my musical knowledge. For the bridge part of Oleo that goes, D7 G7, C7, and F7, would I be right in assuming that you would solo over these chords using the arpeggios of each and the mixolydian of each?

After that what would be the best way of combing the two during improvisation, being that I would have to think a different fingering of each.
It seems like this section might be a turn around trying to get the key back to Bb? Well at least that's what I'm assuming after seeing that. Playing the mixolydian of each will give you a really vanilla sound, but I wouldn't necessarily just play arpeggios. There's a lot more you can do there. Unless what you're aiming for is practice for targeting chord tones?
GoldenGuitar is right, that section is simply back cycling to the Bb chord. So you are basically going around the cycle of fourths until you reach F7 (V7 of Bb).

I mean you could view them as mixolydian if you want, but i would probably rather view them with the scale choices if they were going to their diatonic chords, that gives you some more choices to play with tension.

For example, D7 would go to Gm (vi in Bb), G7 would go to Dm (iii in Bb), C7 would go to F7 (V in Bb) and F7 would go to Bb (I in Bb).

If i were to take a scale approach to those changes and i wanted to add more tension i would try playing these scales.

D7 = D Mixolydian b9b13.
G7 = G Mixoldyian b9b13.
C7 = C Mixolydian.
F7 = F Mixolydian.

On the other hand, using arpeggios with enclosures and approach notes is also a vaild option, and what many bebopers did over such progressions.

A good way to find possibilites how to solo over a tune is to learn a chorus over the tune by a soloist you enjoy. See how they tackled the tune and navigated the progression. Often you will walk away with a new harmonic concept to work into your practice routine, or a new way to view navigating the fundamentals (scales, arpeggios).
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Don't forget the arpeggios lie within scales ... so don't separate them out ... start with a scale fragment into an arpeggio, or vice-versa, for example.

Another nice sound to try is Lydian b7. So, D Lydian b7 (A melodic minor). It's a subtle change from Mixolydian (Lydian b7 has a #4, otherwise same as Mixolydian). Etc.
Everything people are saying above is good, but here's another way of looking at it:
Oleo (the whole tune) is in the key B flat, but the bridge chords aren't in the key so try

D7- B flat #5
G7- B flat #1 b7
C7- B flat Lydian
D7- B flat (or B flat harmonic minor)

It'll give you a slightly different perspective on things.

PS: I know someone is going to say there's no such thing as B flat major #1. I don't care.