Under normal circumstances this would be a good idea, but I would be very hesitant to get a lawyers letter drafted for this matter: it could seriously backfire.
In your initial post you state that you have obtained an arborists report. This could not conceivably have been done without the arborist being able to be very close to the tree to inspect it.
Unless the tree is absolutely on your property line (you don’t actually state this or in fact what issues the tree is causing or may start to cause), the arborist and yourself would technically have had to trespass on your neighbours land to get close enough to actually inspect the tree.
As you have then gone and raised the stakes by sending a lawyers letter, the strata, when they do respond, will need to do so in kind (ie with another lawyers letter)….if you or the arborist trespassed on their land in the first instance, can you see how the exchanges will likely play out?
Strata corporations do not have a community charter and are therefore notoriously slow to deal with. If you really want an answer, ask for the strata managers details from the owner that you have been talking to and then pay a visit to his/her office.