I have not personally tried this, but I believe creating different Groups (in the box shown below) will cause them to get evaluated (and hopefully sent) separately. Please give it a try and let us know if that works or not.
That sounds promising, but my alerts already have different groups. They are the same as my rule names, perhaps due to migration after some upgrade. However, there are many warnings in the logs:
logger=ngalert t=TIME level=warn msg="rule declares one or many reserved labels. Those rules labels will be ignored" labels="alertname=xxx load alert"