The maximum draw won't have a huge impact on your decisions about battery and panel capacity. Instead, the maximum draw imposes a requirement on the inverter you'll use to convert direct current (DC) from the batteries to alternating current (AC) for the house. Assuming you want AC in the house.
If you try to draw more than the maximum the inverter you select is capable of, you'll trip a breaker in the inverter and lose power until you reset it.
To determine the battery capacity, you need to determine how much power you think you will actually use and then decide what proportion of that you want the battery to be able to supply. Then you need to decide on a charging schedule (I will assume 1 full charge per day for the rest of this post). For example, if you use 10kWh/day and you want the battery to supply half of that (with the balance made up by daytime PV production, grid draw, or disrupted service), then you need a battery that can supply 5kWh. (If your power usage took the form of a single instantaneous load, there would be additional factors. I will assume it is relatively evenly spread through a 24 hour power, perhaps with a few peak usage times that are around twice off-peak usage. Note also that taking power from the battery is not 100% efficient so you will actually have less than 5kWh delivered to household loads at the point that you have taken 5kWh from the battery. Factor this in by making the battery slightly larger than you think. Exactly how much larger depends on how efficient you think the system might be. Higher DC voltages tend to be more efficient than lower DC voltages. The inverter will have an efficiency rating. Consider these and come up with a number. 75% efficient might be a reasonable ballpark guess, ignorant of any specifics of what system components you select.)
The maximum regular depth of discharge of a lead-acid battery is 50%. This means that a 10kWh lead-acid battery can supply your 5kWh (50% of the total rated capacity). Lead-acid battery lifetime is greatly extended by reducing depth of discharge. If you wanted to limit yourself to 25% depth of discharge you would need 20kWh (of which 5kWh is 25%).
Once you select a battery capacity, you can size your solar panels. The panels need to be able to replace the power you take from the batteries. If you plan to take 5kWh from the battery each day, you need to replace 5kWh each day. (Again, putting power into the battery is not 100% efficient. You will need to generate more than 5kWh to put 5kWh into your battery.) Next, decide how much power you want to replace from PV and how much you want to replace from the grid. If you select lead-acid, you don't really have the option not to recharge fully once each day. The act of existing at a low state of charge (even with no load) damages lead-acid batteries and reduces their lifespan. You don't want to take a 50% discharge from your battery and then spend a week recharging it. You want it back to 100% as soon as possible (with a daily charge cycle many folks take about 12 hours to get back to 100% charge).
As for battery type, chances are lead-acid is going to be the best choice for you. Lithium-ion is sexy right now but consider its advantages: energy density. If you're building a car, great: pack as much energy in to as little mass as you can. If you're filling a battery shed... well, chances are you're not filling it. You're going to have space left over. And what are the disadvantages of lithium-ion: high total cost of ownership (higher per-Wh cost, lower cycle count than lead-acid). So you can have sexy but you've got to pay for it.
There may be special circumstances where the general reasoning doesn't follow but you haven't given any hints that you're in such a circumstance.
(One interesting case may be that you already own an electric car or two. In that case, you have your battery already. It's in your car. Why buy another one?)
Finally, consider what the point of all this is. You have a grid tie. You don't seem overly concerned with cutting your carbon footprint (you asked whether you should bother with PV at all). You will end up spending a lot of money for this equipment. If you don't go with PV to charge your batteries, all you're doing is introducing inefficiency into the system (well, if you set it up right, you may be gaining backup power for grid failures, too). You said this project is "due to limitations on the street fuse". If that's the case, how do you plan to use the grid to recharge your batteries? Maybe you're trying to use the batteries as a load leveling system? If that's the case, many of the factors involved may be different and a different set of technologies may be more appropriate.