Hello again Emzil.
Now I have the chance to look at your revised information and done some calculations, I make the following recommendations:
You will need a minimum of a 110AH leisure (caravan) type battery to provide the source of power you are looking for.
You will need a 1500W 12Vdc to 230Vac inverter with a "Pure" or "Full" sine wave output - I will explain why below.
I would also suggest you arrange for 50W or bigger Solar panel and charge regulator to help keep the battery charged during the day.
When considering specialist equipment for a commercial venture, it's particularly important to consider it reliability. And for that reason I have rounded up several of the values you have given or are calculated to give a greater margin of operational performance.
I'm sure someone will wonder why I have recommended such a big inverter, when the compressor is only a 1/8HP unit. The reason is electric motors are usually shown with their normal running current or power in this case you have 1/8 HP which is the equivalent to 93W (Watts), This needs a current of about 0.5A (Amps) at 230V. What this doesn't cover is the fact that when ever an electric motor is powered up from standstill, the motor will actually use a lot more current just to get it turning. The current drops back to the steady state value as the speed rises.
The surge current for an unloaded motor is usually assessed at about 4 times the steady state current, but becasue most compressors with auto switches turn on when there is still considerable pressure in the receiver tank, this additional pressure increases the load on the motor which increases its surge current, and I am basing my recommendation on a factor of 10x steady current which is going to be in the order of 5A. at 230V. A simple calculation will tell us the required wattage that represents VxA = W; 230 x 5A = 1150W (Watts) or 1.15kW.
We also need to consider the load the lamps will add, which from your specification is 2 x 45W = 90W so that is a total peak power 1.24kW. The inverter needs to be big enough to cope with this amount of power conversion without stress.
You may well find some inverters are specified as having a 1kW capacity with 1500kW overload margin, and on the face of it might be suitable for you, but I would advise against them, as the peak loading they expect is very infrequent, whereas yours is going to be several times an hour.
Inverters of more than 1kW capacity are typically available in 1 2 and 3kW models. You may be lucky and find a 1.5kW model but in my view that is the minimum size you should be considering.
I have specified a "Full" or "Pure" sine wave inverter becasue the output wave form matches what we get from our domestic mains supplies. Quasi and modified sine waves have nasty transient wave shapes and these reduce the efficiency of motors and transformers. You can sometimes hear motors and transformers buzzing or "singing" on these modified outputs, They usually become hotter, and sometimes wont work at all and will ultimately burn out earlier. So only consider Pure or Full Sine wave output inverters.
There is a trade off for using a pure sine wave inverter, The inverters efficiency drops to something around 67%, so we need to add about 1/3 more to battery capacity to cover for the inverters inefficiency.
So to work out the battery capacity you need, we have to add all the Watt Hour (WH) power demands together for the two days you have specified.:-
90W of lamp for 2 hours per day for 2 days = 360WH
93W Compressor for 4 hours per day for two days = 744WH
Grand total = 1104WH
Add 1/3 to cover inverter losses = 1468wh or more conventionally 1.47 kWH
This will require a 12V battery to be able to supply 122AH (Amp Hours) of capacity. You can get leisure batteries of upto 220AH but these are quite expensive.
However there is an alternative strategy, and that is to use a solar panel in conjunction with a smaller battery. This would have three principal benefits, firstly it will to partially top up the battery during daylight hours. This will means you would almost certainly be able to use a typica 110AH caravan battery, which is smaller and more more manageable. Secondly it will help to ensure the battery's terminal voltage will be above the inverters cut off value, and thirdly It may extend the number of hours you could trade in summer.
A minimum panel of 50W but the bigger the better, again for reliability and longevity of the battery the panel would need a charger regulator.
Another power saver could be looking at LED lighting which could see you lighting power needs drop to about 20W. Don't look at led strings but more at the replacement spotlight GU10 fittings. etc.
I wouldn't recommend using ebay to source an inverter, you would be better going to a canal boat or motor home specialist, again for reliability and spares and service when its needed. This wont be a cheap exercise.
As am alternative solution, You can certainly get 12v LED lights and It would not surprise me if you can't get a suitable 12V compressor . This would negate the need for an expensive inverter, But I would still suggest you consider a solar panel to keep the battery being at least trickle charged.