If you will not uninstall, you will not lost your data and icons. Just install over exists.
As for "Getting back to my first question : is there a way to customize my client's standard Google Map icon on MY map ? (talking about the small pictogram in a circle and the name next to it). Because of course I get both the standard Google Map icon AND my custom icon on top of it. A little messy."
As I understand right - you get your markers with icons, and default Google's POI.
If you want to disable POI, you can use styling maps.
Here description
wiki.zhuk.cc/index.php/Zh_GoogleMap_Description#Styles
There are 2 links there, for description and for wizard.
For information: Google Maps JS API had limit for using styling feature (daily or something else). You need to check does it have limits and if yes - analyze do you fit it.
Today I tried to check info
developers.google.com/maps/pricing-and-plans/#details
And didn't find about styling and limits, probably they can remove it, but you need to check it.