You normally use a combination of a 0.1u ceramic and a larger (4u7 in your case) electrolytic. It's rare to use ceramic capacitors of that value.
That 4u7 apparently is responsible for providing the power to charge up the MOSFET gates. A quick glance at the data sheet gives me an input capacitance of 2.55 nF. You have 16 of those, 41 nF total. A 4u7 cap should be enough to handle that (it's >100 times larger).
Looking at data sheets for other MOSTFET drivers I notice they mention using a 0.1µF ceramic and a 1µF low ESR film capacitor.
MCP1407 data sheet:
4.3 Decoupling Capacitors
Careful layout and decoupling capacitors are highly recommended when using MOSFET drivers. Large currents are required to charge and discharge capacitive loads quickly. For example, 2.25A are needed to charge a 2500 pF load with 18V in 20 ns. To operate the MOSFET driver over a wide frequency range with low supply impedance, a ceramic and a low ESR film capacitor are recommended to be placed in parallel between the driver VDD and the GND. A 1.0 µF low ESR film capacitor and a 0.1 µF ceramic capacitor placed between pins 1, 8 and 4, 5 should be used. These capacitors should be placed close to the driver to minimize circuit board parasitics and provide a local source for the required current.
Would that be advisable in favor of a ceramic or electrolytic 1.0µF capacitor? I have some ECQE4105JF 1 µF polyester capacitors but the ESR value is not listed on the datasheet.
Could you use cheap TTL-level MOSFETs to switch 12V to the 80NF70's?