score:3
Have you been to the 'treinreiswinkel' (in Amsterdam or Leiden) or their site? You can also call them.
They are a very specialist company and at times arrange things they did not have in their standard pack. (And in later years those do sometimes show up on their website.)
For train connections, I prefer the German rail website, which you can set in a lot of languages including Dutch.
Some features I liked have disappeared so you might want to use an other site which does what you want. The Dutch international site has improved so that might be an option. (Also available in English.)
All the French rail sites and the apps have an 'add car' option and from reputation, and comments on your question, I'd say they do what they promise.
This seems their new international version, available in many languages. (Note, if you set it to Dutch you might get the Dutch names for cities like Lille/Rijsel but you can get around that by using the English version.)
There is also Rome2Rio, which I do not like that much but at times it finds solutions other sites ignore.
As I do not have a driving license I leave the renting of cars to my friend/travel companion who does the driving.
Several times we have combined long distance travel with a local car journey by renting from a city location.
Do call ahead to make sure your selected rental location is open at the time you need it.
The solution for awkward timings might be to stay overnight in a hotel near the station you arrive and all cities and most towns have a hotel near the station, often several.
This has the advantage that one of you can stay in the hotel with the gear and child while the other collects the car.
Traveling by train I have seen quite a few rental locations near (main) railway stations, but (going by the comments) they may not always be open when you expect them to.