If the MoM isn't easy to snipe and you are set on doing a single war against Roosevelt "properly," then I see you having a few options:
1. Detouring to Rifling, building Cannons, and Drafting Riflemen
2. Building Cannons, going after a few of Elizabeth's Cities in a short war, and focusing on getting up Factories and Coal Plants before pumping out Military Units to go after Roosevelt
3. Forget about a detour to Rifling, switch into Slavery, and whip Cannons + Cuirassiers
4. Detouring to Rifling, building Cannons, and whipping Cavalry
Personally, I would put less emphasis on angering Elizabeth and just plan to take her land as well as Roosevelt's, so keeping her happy no longer becomes a priority and thus multiple wars with Roosevelt can make sense.
If you want the wars to go smoothly, plan for Rifling. Riflemen are cheaper on your population points, but against Grenadiers or a more backward AI, Cavalry are a more effective choice.
It's hard to say whether you want to invest the extra population points purely in Cavalry, though... if you have The Pyramids and can switch into Police State, then 30 * 1.5 = 45 Hammers per population point (scale the number for your Game Speed) is justifiable, but without Police State, Nationalism and Drafting is a far more efficient use of your population points (110 Hammers or whatever the cost of a Rifleman is per population point).
The main downside with Drafting is that you're looking at only getting 15 Riflemen (possibly scaled to more for a larger-sized map) for 5 turns of Nationalism, so while it's a good population-to-Military-Units conversion, you are limited on how much you can take advantage of it.
It's not a terrible idea to increase the Espionage Slider while in Nationalism, especially since you won't get as many Flasks without Bureaucracy, if you can find a good use for the Espionage Points (City Revolts, stealing Techs, Spreading Culture, stealing a treasury, etc), that is, if you plan to build some Spies.
I'd probably say:
1. Go for Rifling ASAP if you want a smooth war (or multiple smooth wars)
2. Pre-build Cuirassiers and Cannons to be whipped (leave the Cuirassiers in the build queues so that they'll become Cavalry on the turn after learning Rifling)
3. Launch a non-MoM Golden Age to switch into Nationalism + Police State + Slavery, then Draft (don't over-whip Cities before Drafting, as you need I think 7 population points to Draft) and Whip out a good stack of Riflemen, Cannons, and Cavalry
4. Steamroll Roosevelt first then Elizabeth
If you wait a while, you'll just need the next level of tech to have a smooth war (probably Tanks). Basically, you want to first leverage your tech advantage into unlocking efficient Military Units, spam those Military Units, and then get them to the fields quickly before your technological superiority dries up. It'll be a far more efficient approach than, say, waiting to get Rifling in trade from Elizabeth later just to save on Flasks, only to have to spend a ton of Hammers on Tanks to take down her Redcoats.
Going for the MoM after starting a GA still makes sense, as I'm gonna start two further ones. Then the MoM didn't pay off that greatly, but better 1 free GA than nothing, and getting ND is also nice.
A better way to look at it is that for a smooth war, you have to switch Civics and thus you'll save, say, 3 or 4 turns of Anarchy for 4 less turns (maybe scaled by Game Speed) of Golden Age. I'd take that trade any day, as a turn of Anarchy is quite painful compared to a non-Golden-Age turn, with probably the only exception being early in the game when you're bleeding Gold per non-Golden-Age turn (i.e. pre-Currency after a successful hard rush), basically at a time when Anarchy isn't a bad tradeoff due to saving on Commerce in exchange for lost Food and Hammers.