Let me start off by quoting Wikipedia on Guadeloupe:
Guadeloupe, like the other overseas departments, is an integral part of France. As a constituent territory of the European Union and the Eurozone, the euro is its official currency and any European Union citizen is free to settle and work there indefinitely. As an overseas department, however, it is not part of the Schengen Area.
Basically, as a territory1 situated outside of Europe, Schengen does not apply, but as a territory situated in the European Union, free movement applies, allowing2 the scenario you describe. For France, this is the case for the territories of Guadeloupe, Réunion, Martinique, French Guinea, Mayotte3, and Saint Martin4. Other French territories are not generally considered to be part of the European Union5.
This means that French immigration law generally applies to these territories6. For stays of 3 months or less, no formalities are required as guaranteed by Article 6(1) of the Citizens' Rights Directive, other than holding a valid identity card or passport. While stays longer than 3 months can be subject to formalities, it doesn't appear to be the case for a 6 month stay in France (French link). However, if asked, you are required to provide proof of your employment.
- There are specific legal terms at the French and EU levels for different types of territories. To avoid confusion and complexity, I'm sticking with "territory."
- Allows, not guarantees. Your employer might object because of tax ramifications, timezone issues, etc.
- Labour situation currently unstable (French link).
- Unlike the others, Saint Martin is not considered an integral part of France and so expect more than usual amounts of deviation from French/EU law.
- However, France has chosen to extend this EU freedom to the non-EU territories of St. Pierre & Miquelon and Saint Barthélemy (French link).
- No answer on French overseas territories is complete without a final disclaimer that there can be deviations from normal French/EU law, so consult a lawyer if you're concerned (or likely better in this case, a French immigration official). However, I did skim the relevant law wherever Guadeloupe was mentioned and it seemed like all deviations were related to asylum or not being in the Schengen area.