Good luck with this... should be worthwhile. A couple of general comments:

For EPS make sure you compare like with like

• I assume Tesla will be with Gateway for automatic changeover
• Make sure quote for other makes includes capability for automated changeover... for whole home. Not just part of consumer unit etc
• Also that inverter size is capable of max draw. Tesla will likely meet that - I don't know about others. e.g. Start up of heat pump is a surge? 
• Probably requires an earth rod - decent installer will know that! 

Hopefully good tariff options open up. Agile could be worth considering if your battery is big enough and if charge/discharge rates are maximised. 

I have a very similar set up in almost all respects, except that  I do have an ev.  Just a few comments.

1 Eon didn't check that I have an EV when I went to next drive.  Of course they can probably tell from usage pattern, but I doubt they bother.

2 With a Vaillant you can get the OVO heat pump add on, 15p for your heat pump usage.

With my situation I couldn't make the business case for a battery stack up, but without an EV and if the OVO rate doesn't suit (I don't think they do a good export tariff) then it likely does.

i remember hearing about that HP tariff. im wondering how they work out what power the heat pump is using and what the house is using separately?

one installer of PW3's and solar who had suggested the Myenergi, i think due to the cheaper pricing and possible DNO limits not making a bigger inverter useable, is MCS for solar but not for battery storage. what are the implications if i use such an installer? will i have issues going onto some of the tariffs that control the battery for instance?

A few comments:

1  MCS - my solar/battery installer wasn't accredited initially and I couldn't open an export tariff with Octopus until they were.  It took 6 months or so.

2  I just looked at the Ovo HP tariff and they use the cloud account to determine HP power usage.  I have a Mitsi Ecodan which they don't support yet - I am on the waiting list.

3  I believe that it is possible to have import and export tariffs with different suppliers.  Therefore you can have the Octo export at 15p without an import. (I haven't done this yet though so the rules may have changed)

Posted by: @adamk

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wondering how they work out what power the heat pump is using and what the house is using separately

They use the info from the heat pump which they obviously deem good enough.

Posted by: @downfield

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 I believe that it is possible to have import and export tariffs with different suppliers.  Therefore you can have the Octo export at 15p without an import. (I haven't done this yet though so the rules may have changed)

Yes it is but some suppliers will only offer the best export tariffs if you also import from them.  You need to check the small print before doing this 

We have a Tesla Powerwall 3, recently installed, so I maybe biased.

With a Gateway, it will provide full home backup. A separate earth rod must be installed when doing whole home backup, but that is easy enough to do and the installer should be very aware of this. We have tested the full home backup and it switches to the battery seamlessly, no lights flicker, flashing clocks or internet drops. The only way you'd know is the notification on your phone in the App.

The PW3 provides 185A LRA (Locked Rotor Amps) of load start capability, so should in theory have sufficient capacity to meet any startup loads when an ASHP starts up during a grid power cut/outage. I have not tested this yet - maybe that is something I could do tomorrow if anyone is interested

Once you have reasonable battery capacity, a tariff such as Octopus Cosy becomes viable where you can recharge the battery 3 times per day in winter and there is 6h max between cheap slots. Current rates are around 13p for cheap rate. This will be our first winter with a battery, but I'm hoping the 13.5kWh capacity will be sufficient to see us through the most challenging 4pm-10pm period (where usage is heavier and there is no solar to help). Adding sufficient battery capacity to charge once overnight and then last the day is probably not financially viable for heat pump owners, but the Tesla PW system can be expanded by adding up to 3 further 13.5kWh expansion packs.

I think each battery system option provides pretty much the same, but check with individual installers on your final choice that it meets your needs.

If I had to criticise Tesla, it would be the App. It's control is simplistic to say the least - 2 modes and the App uses it's inbuilt algorithms to decide what's best for you. For many folks who like a set and forget system, this will work absolutely fine, but I like fine control and being able to tell the system what I want it to do, and when I want it to do it, so I find this lack of control frustrating.

The only other potential consideration I can think of is that Tesla are now approved for the Octopus Intelligent Flux (IOF) tariff, which can be highly lucrative during summer if you generate more solar PV than you use. They do not currently support Sigenergy or Myenergi Lbbi systems. We use Cosy in winter whilst we are net importing energy and IOF in summer when we are net exporting energy.