Unfortunately there's no good way to do this. The EU regulation does not provide a mechanism for a non-EEA fiancee (or similar) to join an EEA fiancee, nor has the UK done so in its implementing regulations.
This leaves the "normal" immigration rules. However, if you apply for a marriage visitor visa, you may run into difficulty related to the fact that your ultimate plan is to live in the UK for some time. Even if your intention is to leave after marrying, within six months of entering, you may have difficulty establishing that to the satisfaction of the officer evaluating the application.
The marriage settlement visa, on the other hand, is unreasonably expensive, and requiring you to use that visa would be contrary to the spirit, if not the letter, of EU law.
If you can show that your relationship with your fiancee is "lasting," you may be able to qualify for the EEA family permit now. There used to be a relatively strict requirement of two years' cohabitation, but the current guidance (pdf) leaves more wiggle room. Still, it seems likely that you will be unable to qualify. If you can qualify, however, then you won't need to get married to be able to live together in London.
The simplest solution would probably be to get married outside the UK. Some US states, including New York and Nevada, allow nonresidents to marry. Since your girlfriend in French, you may be able to marry in France.
Otherwise, you have already identified the probably best solution, which is to use a marriage visitor visa.
After you're married, you can enter the UK without a visa. You don't need a standard visitor visa. In fact, you shouldn't apply for one because you won't be a "genuine visitor." You could, however, apply for an EEA family permit in the US.
If you enter the UK without a visa, you have the same choice: to present yourself as a (visa exempt) visitor or as the spouse of a UK-resident EEA national (note "national," not "native"). Again, you should take the latter option, since you aren't in fact a "genuine visitor" and trying to pass yourself off as one could complicate matters. You should therefore have the appropriate documents on hand when you arrive; these will be the same documents that you would submit for an EEA family permit application.
The value of applying for the EEA family permit beforehand, therefore, is that it allows you to avoid the stress of finding out at the border whether your documents will be accepted. On the other hand, it seems (anecdotally, from postings on this site) that border officers may be more likely to consider your claim favorably than are visa officers.
There's another option you haven't suggested, which is to enter as a marriage visitor and then apply for the residence card without leaving first. This is actually legal because once you're married you fall under the EEA regulations, and any immigration restrictions under the "regular" immigration law are without effect for someone who has a right to reside under the regulations. But this is precisely why your marriage visitor visa may not be granted: you will have difficulty showing your intention to leave the country before the visa expires.
Finally, be very careful about saying things like that you "want to get married so we can live together while she is working in London." This could cause trouble under the definition of "marriage of convenience," which says:
“marriage of convenience” includes a marriage entered into for the purpose of using these Regulations, or any other right conferred by the EU Treaties, as a means to circumvent—
(a) immigration rules applying to non-EEA nationals (such as any applicable requirement under the 1971 Act to have leave to enter or remain in the United Kingdom); or
(b) any other criteria that the party to the marriage of convenience would otherwise have to meet in order to enjoy a right to reside under these Regulations or the EU Treaties;
This means that if your main reason for marrying is to be able to move to the UK, you cannot use the marriage as a basis for moving to the UK. You might think that this shouldn't apply to people who were in love anyway, where the immigration benefit wasn't the only reason but merely the immediate motivation for the decision to marry. However, there have been cases where the Home Office has in fact interpreted this as applying to such couples.