Under the circumstances you describe, you won't be working in France, so you won't need a French work permit. But French payroll systems will generally assume that the worker is in France, so subject to French taxation, labor law, and so on. You won't be subject to those things. If the French company has no payroll in Croatia, they won't be able to comply with Croatia's laws concerning taxation and so on. In short, they can't hire you as an employee.
The usual way to do this, therefore, is to set yourself up as a consultant.
If the company is telling you that you need a French work permit then they apparently have no experience with this. There isn't much you can do unless you can somehow convince them that they are wrong about the work permit, or at least that hiring you as a contractor would obviate the need for a work permit. If you can't convince them to let you bill them as an independent service provider in a business-to-business transaction, you probably aren't going to be able to work for them.
I don't know anything about Croatia's law concerning sole proprietorship. In general, you may be able to create a corporate legal entity that would bill the French company and pay you a salary, or you may be able to operate a sole proprietorship without creating or registering anything, or you may be required to register without being required to create a legal entity. Different countries have different systems. In countries where you aren't required to create a corporate entity, you may or may not have the option to do so anyway, which can bring some benefits around taxation and liability, at least in some countries.
In any event, you'll need to figure out the rules governing the collection of VAT for intra-EU cross-border services, and your legal entity or you will no doubt have to register with Croatia's revenue authority for the purpose of remitting the VAT.