Should lime and P be applied at the same time?
No matter how phosphorus (P) is applied or when it is applied it will be tied up with something. A soil pH of 6.5 is the very best one can do to reduce this fixation of P.
If fertilizer P is applied below a pH of 6.5, especially below 6.0, more of it will be tied up with aluminium and iron in the soil. So not applying lime would be bad also. If the pH is above 7 then more of the P will be tied up by calcium in the form of insoluble calcium phosphates (rock phosphate). So, you cannot win either way.
If you have a low pH, say in the high 5’s, and you apply lime and you add P also. You would have to get the soil pH where the lime is placed above 7 for much of this additional fixation to occur and it is going to be difficult for this to happen since you are starting at a low pH and lime is a rock and is only slowly soluble.
Bottom line is do not worry about P fixation and keep the soil pH about 6.5. Under the best of circumstances, you are only going to get about 30 percent of the P you apply in the crop. Very small differences in P fixation would happen with or without the lime.