Which statistical test to compare weekly weight changes? (Gain or loss)

Neither a t-test nor ANOVA is appropriate here.

1. Your measurements at different times are not independent
2. You have a continuous covariate--time.

One approach would be OLS regression. It is possible that a mixed-model would be even more appropriate.

Suppose you are measuring the value of variable $W$ representing the weight. In addition, you have an ID for each subject $ID$. You likely also have some additional covariates such as sex $S$, diet $D$, breed $B$, etc. Finally, you have the time-stamp of each measurement of $T$.

Forget which week and how many times during the week. I'm assuming that it is the raw effect of time you are interested in. So just put a time stamp on each measurement. It won't matter if some measurements are skipped. The time stamp will still keep them in the correct order.

You could then perform the regression

W ~ T + S + D + B + ID

and any interactions you think might be involved.

This tutorial leads you through the steps of a repeated-measures design analysis, along with post-hoc analysis (don't forget the post-hoc!).

https://rcompanion.org/handbook/I_09.html

Many might argue that a mixed-model approach would be superior, with the subject ID as the random factor. In a simple before/after study you assume the effect of the treatment (if any exists) has occurred by the time you take the "after" measurement. However, the rate at which different subjects respond might be different although in the same trend. Treating this trend as a random effect increases your poser.

You can read about these types of models here:

https://www.jmp.com/en_dk/learning-library/mixed-models.html

https://www.stat.ncsu.edu/people/davidian/courses/st790/notes/chap3.pdf

Additional Note:

This question just popped up on the main feed:

Difference between Repeated measures ANOVA, ANCOVA and Linear mixed effects model

You likely need a paired t-test here, which is appropriate when you have paired (dependent) data (e.g., before/after some new drug was administered, before/after some new diet regimen). t-tests are appropriate for small samples, so I believe this will suit you well.