Hi!
Some personal thoughts/opinions: integration is about finding the similar cell types/states across data sets (either with or without batch effect).
An experimental batch corresponds to a set of samples that were processed simultaneously in the same manner and, thus, reducing the effect of technical artifacts/noise.
As I see your experiment, you've two biological conditions plus control where you want to assess/study the effect of two treatments on the same batch cells in the course of time (longitudinal experiment).
Thus I don't agree with:
there is most certainly a batch effect associated with days of collection of samples.
Unless you collect the samples from the different conditions you have in different days for the same timepoints, e.g., by starting conditions at different times, I don't see how the collection across the distinct time points could be considered a batch or introduce noise. If any (noise), should be affecting equally the different conditions (this is my opinion).
From your plots above you can clearly see that perhaps one of the biggest difference seems to be between the sample NT-D0 (which I guess refers to the original batch cells) and the remaining cells/conditions. Regarding the uncorrected/unintegrated projections, you can see that cells are being projected slightly based on condition, as I guess you would expect. I've problems identifying the control cells as they seem to be quite spread out.
All this said, let me answer your questions:
1. Is our experiment utterly borked or Seurat is too zealous in its batch-correction?
I don't think so. The CCA method in Seurat tends to prioritize batch correction over bio-conservation (see Luecken et al., 2022) which might be recommended for integration tasks where the batch is stronger or data sets more difficult to integrate as cross-species integration tasks.
Seurat documentation provides an alternative method based on RPCA (Reciprocal PCA) which prioritizes bio-conservation over batch correction (see documentation).
In case you think the CCA is being to "aggressive" over-integrating, you might try the RPCA method.
2. Should I even apply the batch correction given the fact that these aren't the same cells that come from different batches, but (supposedly) phenotypically different cells?
It depends on your goals. If you want to find the "shared cell types/states" across conditions you might want to integrate. In this case, I think you know the identity of the batch cells and, therefore, you might not want to integrate, but instead see how cells change with treatments throughout time. If the latter, I personally wouldn't integrate the data as this would mask the biological differences between treatments.
The only point that I see where integration could be beneficial is if in your batch cells you know that you've multiple/distinct cell types, such as in PBMCs, and you would like to integrate the different cell types to see how treatments affect the different cell types.
Discuss some of these points with your collaborators and look for other answers in the forum before taking a supported decision.
I hope this helps!
Best,
António