Electronic – use higher rated capacitor and larger package (for capacitors, resistors, etc?)

Yes, you can do that.

To be on the safe side of things I suggest an addition:

Put two 50KOhm to 100kOhm resistors in parallel to the capacitors. These resistors make sure that:

• The voltage level at the junction between the capacitors is close to 1/2 of the total voltage.

With ideal capacitors the junction would be at 1/2 of the total voltage. The world isn't ideal though and you will get capacitors that have tolerances in capacitance and the internal series resistance. When you charge them they will in practice get a voltage that is somewhere else but not exactly at 1/2 of the total voltage.

I suggest you try this out at home using low voltage (12V or so) and two different 100µF capacitors. You may be surprised how far off the voltage after a charge cycle is.

Adding the resistor voltage divider gives the capacitors a voltage reference.

• The capacitors discharge over time making your device a bit less dangerous.

Capacitors of that high capacitance and voltage can easily kill you if you touch them. They also hold voltage for quite a long time. Worse: Even after discharging, the capacitors may re-charge on their own due to an effect called dielectric absorption.

Having a resistor in parallel to the capacitors keeps that somewhat in check.

Last word of warning repeated:

The charge stored in the capacitors can easily kill people. If you don't yet know what you're dealing with please carefully read safety rules from the DIY tube-amp community. They deal with with that stuff each day.

Edit: Since the OP asked why such a circuit can kill people, even if it's powered just by a 1.5V battery:

Your disposable flash charger is a circuit that transforms the 1.5V up to some much higher voltage at a much lower current. This current is used to charge up the capacitor. Charging takes a while because the charging current is low, but once the capacitor is charged the energy can be let free instantly and currents of multiple amperes are possible.

Now what happens if you put your fingers across the leads of a 330µF capacitor loaded with 300V?

First thing is, that current starts to flow through your skin. The skin resistance is somewhere between 1KOhm and 100KOhm.

Lets say it's a hot summer day and your skin resistance is on the lower side of things. 10KOhm let's say. You'll get a shock, but this will likely not kill you because the current is not high enough yet. 300V at 10KOhm gives 30mA.

However, something else will likely happen: You get burn marks at the point where the current enters your body. And this is critical: Suddenly the high skin resistance is partly gone and your flesh is in direct contact with the voltage. The resistance will drop down to 1000 to 500 Ohm now.

Part of the energy stored in the capacitor will be consumed now and the voltage dropped down a bit. Let's say it's down to 280V now and your body resistance is at 500 Ohm. How much current will flow? 560mA. OUTCH!

There are different sources how much current is required to kill. It also depends on a lot of factors and differs from person to person. A number that I've picked up on the Internet was 300mA for DC currents.

The capacitor will now rapidly discharge through your body and the current will be down in the safer region after half a second or so.

Will that kill you? The answer is: Maybe. You only got one single discharge cycle, not a prolonged exposure to the current. This is good. If the current passes your heart (easy to do: Just make contact with both hands) the chances are quite high that your heart gets out of rhythm. Have bad luck and you'll drop dead. If you touched the capacitor with a single finger the chances are much lower.

That is by the way the reason why you're often advised to put one hand into your pockets if you're poking around in anything with high voltages. It prevents having current flow through the heart.

So best case it will hurt a lot. If worse comes to worse you'll drop dead from from a 1.5V battery.

The bad caps are both 6.3 V, 4700 µF and i was thinking of replacing them with 16 V, 4700 µF

Yes, you can pick a higher voltage without problems.

but getting the better quality 105 °C rated one as the others ones are 85 °C rated.

Again, yes, 105°C will last longer.

HOWEVER:

These caps are most likely at the output of a switching power supply (please check). In this case, you will have to use low-ESR (or low-Z) models rated for this use.

Consider Panasonic FC-FM-FR, Rubycon ZL, for example, but do not use "general purpose" caps in a low-Z position.

Also, please make sure the capacitors fit in the holes. Don't connect them with wires or stuff like that, as the extra inductance could increase HF ripple from your switching supply.