The biggest problem I see is your diplomatic situation. You aren't big enough to treat AE as a high score list and I'm curious if you have any allies? In that situation I'd want one or two strong allies who would mostly be there to dissuade coalitions and other potential attacks. I can imagine a scenario where you launch the great war against France and Russia and Persia immediately see an opportunity to attack you. Do you have DIplo and Influence ideas? When you can stack up both a few AE reductions and some Improve Relations you'll find you can really burn off AE quite fast. When it ticks down at a rate of 5 per year things start to look very manageable on that front.
In terms of being able to conquer and core fast enough it isn't even an issue. You will see the power of 60 - 70 admin efficiency and be amazed. Even more so if you stack Diplo ideas and being Revolutionary and you're looking at a 70% reduction on war score cost for provinces. You can either use that to take 250 OE worth of crap from someone and either make Client States, (I made a Client Stat the other day and force fed it 200 OE worth of land, just kept a stack nearby to kill their rebels and they were fine) release vassals, or just tank the OE yourself. That can be a bit scary but doable if you core quickly and/or have Humanist ideas and are willing to squash some rebels. You can also just use the discount to be able to take 100 OE worth of land for a lower amount of war score. Getting 40 - 50 WS is a lot quicker and easier than getting 90 - 100, and the truce will be substantially shorter.
The other thing you can do - if that coalition is just a few nations right now - attack it, beat it to a pulp, and take 100 WS worth of stuff that doesn't cause OE just to get a 15 year truce and make yourself safer to attack others. Luckily it looks like your biggest AE problems are to the East while the land you need is to the West. Persia aren't going to care that much if you invade Germany and France, so if you can get them placated or neutered they probably won't be an issue in the future.