Chris Wilson, you may want to do an energy/power audit of your existing 1ph loads, including any anticipated future loads (as you just indicated). Other options often become clearer when you fully appreciate what you do with your power, and when, and whether that default useage profile can be not too inconveniently modified during an extended power outage period. Also any thought of deploying a genset option also requires a very good understanding of your load profile.
Obviously 3-ph gensets prefer balanced ph loads, and there are PQ issues for imbalance and short duration high power loads, and diesel maintenance implications for imbalance and for low load operation (ie. overnight operation). And if the system is only used for power outages then you still have a cost/inconvenience to maintain the genset and auxiliary systems for what may be long durations without an outage (given most gensets have a minimum runtime recommendation with a minimum load and you may want to do monthly testing). You may also have an adverse environment that could require cold starting, which has an adverse affect on genset maintenance and start-battery capability, and perhaps fuel type. A can-of-worms .....
100% this.
I use to run small diesel generators on small film sets and use to own one also for film work.
The advice is always to run a 3ph generator as balanced as possible. I have talked to MEMS about it and they say you should always do your best to balance the load but they run generators the size of shipping containers and they are connected together on larger operations...
That being said, does it really damage the generator? Probably not that much if any.
3ph generators always sound better than single-phase when balanced as (and correct me if I am wrong);
A, they are typically larger
B, they have more current phases throughout the rotation of the stater/camshaft which results in a much smoother running engine.
If B is correct, you will probably find that a 3ph under a balanced load will last a lot longer than a single-phase under a typical load because of the lower vibrations, smoother operation etc.
You should also see a much nicer voltage waveform on 3ph compared to single.
If you have ever looked at one on a single-phase generator, you will see why people recommend not to plugin IT equipment into them and that you need an inverter generator for those instead.
A big mistake people make when using a diesel generator is not having a load test check every few months.
In mission-critical deployments such as data centres, government builds and hospitals etc, tests are carried out monthly under load for 30+ minutes.
Backup generators for offices blocks, industrial sites, food factories, maybe 2-4 times a year depending on their policy.
Some people/firms don't do it and then wonder why the backup power does not work when they need it most...
If you are serious about plumbing a backup generator into your house and don't want to over-spec your 3ph generator to suit 100% house load on a single phase, consider splitting your consumer unit (breaker board for the yanks) in two.
Mission-critical and everything else.
Mission-critical having a switchover to either run on mains or backup. These should be things like lights, networking, certain sockets, a shower if you must etc.
As others have suggested, a 3ph board is an option. I would counter that it is actually not a great idea unless if the house is big enough to warrant it or can balance the load well enough to make it worth doing.
If you put the heating on ph1, kitchen on ph2 and everything else on ph3. Well, it could be summer and you are not using the heating so ph1 is no longer being used, though during winter it is 100%.
The kitchen is only used 1-2 hours a day and spikes a few times a day for tea. etc etc
You might actually do better on a larger single-phase generator that can ramp to meet demand unless vs doing all the work to split into 3ph or load a single phase on a 3ph generator?
You need to ask yourself this. In a power outage. Do you really need to have a backup power system in place that can handle 100% capacity or can you live with the essentials?
Don't get me wrong, I love my tumble dryer but during a power outage my priorities shift to either fixing the issue or getting the DNO to fix it

I can live without soft towels after my monthly shower but not my computer or internet or fridge or air fryer...
15kVA is something like 10kW? On that, you can run a 3-bed house with all the LED lights on, a TV, Computer and even a 2kW plugin oven or kettle with ease.
You probably won't be running a decent electric shower though depending on the shower's rating obviously. Electric heating or all the 7.2kW induction hobs? Forget it!
TL;DR get an expert in who specialises in backup generators and get them to work out a plan for you.
My personal option for things like this. Just get a portable inverter generator that can power the essentials in your house unless if power outages are common in your area.
If you need a 3ph for the workshop, consider a static phase converter? They hold their value and running costs would be lower surely?