I'd just update the address everywhere. The person you'll be talking to knows that address changes take time to percolate through the system...And here is what is on my mind, lets say I update address literally everywhere including CIC, now that my app is 'In Process' and based on averages my test invite could be in 1 month. And my updated license might come only in 4-6 weeks making me to show either my old one with old address not matching with that of updated ecas.
Or is it just me overthinking it and I should just go ahead and do the address change everywhere?
If you update the address for your licence online you may be able to print the confirmation, so you've got that to take with you. If you go to a ServiceOntario (or equivalent for your province) they'll give you a confirmation form, so you'd have something in hand that way, too.