Yes, you actually can! There are a couple of options for the Betaflight firmware which are mostly plug and play.
OpenTX Lua Scripts
Radios running OpenTX can use scripts written in Lua to control these parameters with the radio's scroll wheel and buttons. Please see @PaulKendall's answer for an explanation of how this method works.
If you have an FPV setup on your quad, you can make use of the Betaflight OSD to tweak the vast majority of settings you would need access to during a flight session, like the PID values and rate profiles you mentioned. The OSD makes use of the sticks on your transmitter to navigate the menus, which appear as an overlay on top of the camera feed from the quad.
As described in this article by Oscar Liang, the stick combination mid throttle, yaw left, pitch forward, roll centered is used to enter the menu, which should look something like this:
From here, pitch up/down is used to navigate up and down in the current menu, and roll right to select an option or enter a sub-menu. (items with arrows on the right side are sub-menus) The roll axis is used to change a value up/down.
I know this sounds confusing, but it makes a lot more sense once you try it out and gain experience working with it. Remember to select the
SAVE REBOOT after making a change and before going back to flying, or your change likely won't take effect!
Speedy Bee Smartphone App
If you want an experience that is closest to using the desktop Betaflight configurator or don't have an FPV setup on your quad to use, the Speedy Bee smartphone app for iOS and Android may be a good option. You can connect to the quad either using Bluetooth (you must have a Bluetooth module already installed on your quad), or over USB (USB OTG mode is only supported on Android).
Because this app is developed and maintained by a third-party and not the Betaflight project, some new settings aren't likely to be accessible with this app, but it should work for most common settings. I suggest watching Joshua Bardwell's video about this app for more details.