My preferred strategy. But only because of where we live.
Have done this for 3 instances (MHI x 2, Fujitsu x1) in the previous 5 years plus once used a specialist (Daikin) to supply and install.
For anyone in a big city with numerous companies specialising in air conditioning systems, it may be competitive to shop around for a one stop supply and install from a local specialist. It is worth noting the specialist installers usually have a preferred brand and likely price advantage with that brand for their area.
We did one install two years back and another this time last year. In both these instances we managed to purchase MHI splits. Choice reviews are very helpful. Also scored due to timing discounted pricing plus MHI cash backs. The Goodguys in one instance and surprisingly for the other on a hot tip from the MHI distributor rep, Bunnings. We priced the same units including supply and install from a number of suppliers and specialists to confirm we had the best prices.
As none of the suppliers were local 30-50km distant we paid for delivery in one instance and picked up from Bunnings in the other.
We have several local electrical contractors who have refrigeration licenses (dual trade). The difference in quotes between them has been minimal. Also easy to obtain reliable quotes as there is minimal travel required for them to check the job out. We have used our locals both times. In the instance of warranty, if it is an install fault they fix. If it is a unit fault usually they charge us only time to drop in. $100 call out for a 2km drive on the way past to home seems a little rich. We have never needed to make the next call to the retailer for an aircon, so that remains untested. Although the further you are from their nearest service agent the more likely they are to talk to your installing electrician and do a deal. Have experience of that first hand on other items.
It is worth considering any new install will require electrical cabling and work in the household meter box. Getting the power cables to the air con location can be a big part of the labour cost and challenging for some homes. It is worth noting some specialists and the retailers such as HN, the GG etc quote a standard install price (with a list of exclusions/variations). Their install quotes always seemed a little cheaper. We were not game enough to dip that toe in the water.