If it's your first time as Ottos, I'd play them straight, just to get a sense of their unique government, which you lose after conversion.
A few things to be aware of before conversion, especially if you've never played Ottos before.:
1. Make sure you take the Move Capital to Constantinople Decision. This should be one of your opening moves anyways. Don't core Constantiple, the decision will do it for you.
2. Before conversion, consider getting the provinces for the "Sultan becomes Khalifa" event (Cairo, Alexandria, Damascus, and Jerusalem). It's
400 ADM, 25 prestige, & 25 legitimacy (and 25 piety, which doesn't matter after conversion...) . It'll take more time before you convert, but the event is only available as a Muslim. It has a 36 month MTTH, so it tends to fire pretty fast after geting the required provinces.
3. Look at the Coptic events - there are a couple that give increased conversion % for states once you control them and prestige. After you convert, these will immediately fire if you already control these areas. If the rebels don't convert these provinces, you can use missionaries (and maybe an Inquisitor adviser) following the country conversion to do so quickly. You might also want to avoid conquering them until you're ready (except many overlap with the Khalifa decision) so you don't "waste" the bonus by being unable to convert them all in time.
4. You lose the unique government after conversion. This gives you a choice among 3 heirs via event, one of which is usually pretty good. Make sure you milk this (via disinherit if they're all bad) to get a great heir prior to conversion. As a Christian, you'll have to disinherit much more often to ensure good rulers.
Give them dhimmi, later remove one dhimmi from Orthodox and later Coptic one. They will revolt. Take away your armies to Europa and don't supply castles. Sit comfortably on your chair and wait until rebels occupies much enough territories to convert Coptic.
If you have Cossacks, the Dhimmi trick is easiest. When you revoke land from a disloyal estate, rebels pop up. No need to give them an Orthodox province, though - you can make them disloyal by using the Recruit and Tax interactions. Conquer 3-4 of the Coptic provinces east of Anatolia, assign all to dhimmi and revoke all. This gives you 3-4 rebel stacks which should convert provinces faster.
Without Cossacks, assign a missionary and set missionary maintenance to 0. The 0 maintenance is so you don't actually convert the province (or waste ducats trying...). The missionary should set the rebels to Coptic and with the added unrest they'll spawn eventually.
Every province they occupy will convert to Coptic. Religious rebels tend to focus on provinces with a missionary, so you can assign some missionaries to direct them.