As said, the bottleneck nature of Perth makes it a pivotal hub for all services, so 3 platforms would be pushing the capacity barrier currently. In my opinion, Perth merits atleast 6 platforms (2 through for the HML, 2 termini and 2 through for Dundee northwards).
Given that current services from Inverness are being increased to a regular hourly service from Perth, you're talking at least 2 platforms for the Highland Mainline, taking both The Chieftain and Sleeper into account. Also, the service increase from Dundee to compensate for the deduction in station calls from Intercity services would see peak time capacity through platforms 1 and 2 as high as 3tph. Including the Edinburgh semi-fast terminators and eventual scope for a Glasgow semi-fast terminator, as well as initial plans for VTEC to start their morning Stirling service to Perth, at least 2 termini platforms would be sufficient to cope with that.
The layout of the railway in and around Perth could easily hold an advantage. During the reconstruction of the station, it wouldn't necessarily be hard to contract temporary platforms either at the Friarton or next to the Dunkeld Road at Inveralmond, which would at least see Perth still providing a major connection for onward journeys - otherwise you're looking at services running as a replacement bus between Dunkeld/Dundee and Gleneagles.
As for diversions via Leuchars and Falkirk, this was already done on a couple of days last year so I doubt there won't be much standing in the way in terms of capacity, although I might be wrong given the projected service increase on both E-G and Intercity routes.