What is likely to happen, if anything, if I try and export to the grid and import from the grid simultaneously?

Yes, £0.07 23:30 to 05:30 and £0.27 05:30 to 23:30. I can export at £0.15. The grid is just what we call the national electricity infrastructure here. The battery I have has the capacity to support the house between 05:30 to 23:30 with charge still remaining. So I basically pay £0.07 for all my electric. Any excess I can sell back at £0.08 profit (I expect to be a millionaire any day now ). However, once late autumn and winter roll around my ASHP will consume a lot more and I expect to pull on the grid at the higher price. 

I worked for a company that designed and manufactured a wide range of power inverter (up to 1 MW).  We would routinely do exactly what you are taking about.  We would power our inverter from the grid and generate power, pushing it back onto the grid.  We would only be charged for the loses in the system (based on efficiency and any other losses). Again, as others have mentioned, you need make sure that no limits are exceeded (wire gauges and fuses, etc.).

It’s running now and all seems fine. I forgot to take the 8kW inverter into account so judging from what the control app is telling me, I’m pushing the full 8kW with the car taking 7kW and the grid 1kW. The export limit I have is 3.6KW so the inverter would normally limit to that when the load is low (nights I’m not charging for example)

To close this off, everything worked fine - car charged up, battery discharged and then charged backup.  No fiery deaths occurred.  What I’d forgotten about was the 8kW inverter - it can discharge up to that rate although if the discharge is to the grid it limits that to 3.68kW.  Last night with the car pulling 7kW, this initially came from the battery which was also then able to send 1kW back to the grid.  Once the battery had discharged all power came from the grid.

The underlying complications are that:

• I don’t control when the car charges.  I plug it in, my utility company recognises that and sets a charging schedule based on my desired SoC, ready-by time and expected grid load.  This can change as time goes on and grid load changes.
• The battery configuration is awkward.  I can’t amend a configuration, it has to be deleted and then re-created, so it’s a pain to reconfigure it to not export to the grid whilst the car is plugged in, then change it back again.

I’m hoping to get more control over this with Home Automation but I’m away off that at the moment, so the fact I can just leave the battery configured to export then recharge makes life easier even if I lose out on the pennies.

Your £0.15 rate is cheap! I pay about $0.15/Kw flat + $8.00 service fee, But he, he, I don't sell we trade Kw used == Kw supplied!!  So if I use 2kw and give them 2.kw its free ( plus the service fee which I have to pay..)  but sadly I live in an Apartment currently so I am currently looking for something in my service area! 

The standard rate for import between 05:30 and 23:30 is £0.277 so I’m pretty much getting electric for 25% of that.  I also have to pay something like £0.47 a day as a service charge.  You may think £0.15 for export is cheap but it’s about the best on the market - some utility companies are paying as little as £0.01!  
Electricity in the UK is very expensive compared to the rest of Europe (the world) because its price is pegged to the price of coal and gas plus a surcharge for green investment.

I am kind of joining the party late and because I don't have an EV, the post and responses initially came across as a very unfamiliar George Jetson kind of world.  For instance I had not even heard of a house battery.  I am wondering one thing though... if someone didn't have an EV... is the setup scalable.... can someone have multiple house batteries which they charge up during the cheap time and sell back at a higher rate later in the day?

How long does your battery last with that kind of activity, and how much does it cost to replace it when it wears out? I wouldn't calculate "profit" without knowing depreciation expenses...

It's guaranteed for 15 years to be 80% usable.  And I'm not sure what you mean by "that type of activity" - it's a battery that is meant to be charged up and then discharged over time.  None of this is particularly stressful or beyond what it has been designed for.  The whole system is expected to have a payback period of 7 - 8 years, although obviously that is predicated on electricity prices staying the same or going up.  There's a reasonable probability they will come down over that sort of time period.  

Technically, yes, but inefficiencies nab you. I suspect that if it were a good deal more and more would do it, thereby balancing everything out and then it becomes more and more uneconomical, especially if buying a new battery just to do this.

I've heard of a few using old EV batteries for this sort of thing; especially EV owners. That is, when the EV needs battery replacement to improve performance & efficiency they take the old battery and place in the home etc. Then it's a no-brainer as you've already paid for the battery.