One thing to check if you haven't already is to make sure the layer IDs in the Pro publishing document are consistent. In the Map Properties - General there is a setting to "Allow assignment of unique numeric IDs for sharing web layers".
With that checked on you can now manually control the IDs for each of the layers in their General Properties. Look for the "Layer ID" setting.
When published the IDs you set for the layers in Pro will appear to the right of the layer name in REST:
If you don't control this setting the layers will be ID-ed automatically by Pro. I would think each GP output would result in a different ID number.
I've overwritten many services and generally if you keep the layer IDs consistent they will work. Though I haven't done a ton of this with the dashboards so perhaps there is something odd with them.
In my experience an overwrite does not change the Service Item ID used within AGOL. As mentioned schema changes are another story!