In a different topic @dale.roberts provided a Jupyter notebook that utilised an Iris function guess_bounds that might be useful in your case:
According to the ESMF docs
- First-order conservative: Preserves the integral of the source field across the regridding. For this method, weight calculation is based on the ratio of source cell area overlapped with the corresponding destination cell area. If the user areas option (see below) is not used, then the areas used in this calculation are those calculated by ESMF and thus the ones for which the conservation holds. The user areas option allows the user to adjust the interpolation weights so that conservation is based on user-supplied areas.
Which sounds like it should sum when regridding. Easy enough to check.