Generally speaking, it's not possible for people who don't need a short-stay visa to get one, that's not how Schengen visas work (unlike the UK or US for example, where you can apply for a visa for various reasons even if your citizenship qualifies you for the Visa Waiver Program or you're not a visa national). So the French consulate should not give you a regular uniform Schengen visa.
Most importantly, if you would in fact get such a visa, it would not solve your problem. The reason for that is that a Schengen visa does not open any right to stay longer than 90 days in the Schengen area, all visitors are bound by the 90/180 days short-stay limit in any case, even across multiple visas. This maximum stay rule is defined at the beginning of article 5(1) of the Schengen Borders code, independently of the visa requirement defined at article 5(1)(b).
Again in contrast with US or UK practice, the maximum length of stay allowed is not decided at the border by an immigration officer but entirely defined by the regulations (and, for people who need one, the visa). There is no notion of being “admitted until” and no stamp to that effect, only a date of entry and a strict rule border guards are supposed to enforce. So, visa or no visa, you could and should be refused entry if you intend to stay too long or fined on the way out if you managed to enter and overstay.
In fact, not having previously stayed more than 90 days in any 180 day period (and not intending to do so) is also a requirement for the issuance of a Schengen visa, per article 21(1) of the Schengen Visa code, which refers back to article 5(1) of the Schengen Borders code. So even if your citizenship did not make applying for a Schengen visa unnecessary, the logical decision would be to refuse it, since you do not meet the requirements precisely because you want to stay more than 90 days in total. Having stayed too long is also one of the reasons on the standard refusal form.
What the consulate could in theory do is issue a “limited territorial validity” visa, valid only for France. I have never heard of it for a case like that, these visas are rather uncommon and primarily intended to be used for humanitarian or national interest reasons but legally speaking, it does allow a state to exempt someone from the 90-day limit. Or they could perhaps issue a national long-stay visa (it's not a Schengen visa, which means the fees, application form and requirements are different).
In any case, you need to consult with the consulate because if you just submit a regular Schengen visa application without explanation, you run a high risk of a rejection (on the basis of your citizenship) or refusal (because of your previous stay). You should also make sure to read all the requirements carefully, if you are not used to visa applications. For example, people who require a visa to enter the Schengen area must show they have appropriate health insurance for the period of their stay, which is not required for visa-free visits.