... This is a problem, there is no trivial solution, just because you cannot and must not solve both problems (the interpolation problem and your problem of interest) separately: you must solve them jointly.

However, if adapting the algorithm of interest to uneven spaced times or solving both problem jointly is too difficult, you may consider resorting to "Poincaré-Jaynes-Bretthorst" interpolation that can be easily adapted to handle uneven spaced times. Please see my question

uneven spaced time series

for references.

"Poincaré-Jaynes-Bretthorst" interpolation is in some extent exact: it is necessary (according to Poincaré) and also sufficient, provided that you equip yourself with Jaynes' Principle of Maximum Entropy. Essentially, you "just" need to choose the order of the derivative to constrain (that makes no big difference in some cases.)

... This is a problem, there is no trivial solution, just because you cannot and must not solve both problems (the interpolation problem and your problem of interest) separately: you must solve them jointly.

However, if adapting the algorithm of interest to uneven spaced times or solving both problem jointly is too difficult, you may consider resorting to "Poincaré-Jaynes-Bretthorst" interpolation that can be easily adapted to handle uneven spaced times. Please see my question

uneven spaced time series

for references.

"Poincaré-Jaynes-Bretthorst" interpolation is in some extent exact: it is necessary (according to Poincaré) and also sufficient, provided that you equip yourself with Jaynes' Principle of Maximum Entropy. Essentially, you "just" need to choose the order of the derivative to constrain (that makes no big difference in some cases.)