Six weeks out is the right point to confirm the basics are either already in place or booked. Check your pet's microchip is ISO-compliant and already implanted; if it isn't, this has to happen before the rabies vaccination, not after, or the vaccination won't count toward the EU travel requirements.
If your pet's rabies vaccination needs renewing, or has never been given, book it now — a first (primary) vaccination needs a 21-day wait before travel, and six weeks gives comfortable margin if the appointment slips or a booster is needed first. If you're an EU resident without a pet passport yet, this is also the point to book the appointment to have one issued.
If your dog is a breed or type that might fall under France's category 2 dangerous dog rules, start the permit process now. The permis de détention requires a seven-hour training course and, for dogs over eight months old, a vet's behavioural evaluation — both of which take real lead time to schedule, and this is the step most likely to be underestimated.