It's not the capacitor which is limiting the upper frequency. It only limits the maximum on time, which is proportional to the capacitor size.
The 10k pull-up resistor is limiting the current to the MOSFET gate, which acts as a capacitor. The extra RC circuit, slows down the switching speed.
Another transistor is required to quickly turn the MOSFET on.
I'd recommend using a high side MOSFET driver IC, which will have other features such as undervoltage lockout, which will prevent turn the MOSFET off if the voltage on the bootstrap capacitor is too low to keep it hard on. Without undervoltage lockout, the MOSFET will get hot and overheat, if the voltage on the bootstrap capacitor is low and the driver keeps it on, as it will be in the linear region, with a high current and volt drop across it. There are plenty of high side driver ICs. Here's one I found by doing a quick Google for high side MOSFET driver.
IR2127PBF
https://docs-emea.rs-online.com/webdocs/0791/0900766b807910ad.pdf