Josh,
You may need to play around more to get it all set up, but there are a few things that might address your issues.
Note that the RX10IV did add a My Menu custom page in the main menu (the *star* icon)...that will allow you to save 2 pages of your most used menu items, which will greatly simplify having to dig through many pages of menus. You also have the Fn menu that can save 12 control settings for quick access too - between those two, you should be able to save just about anything you'd need to access without having to deep-menu dive.
On recalling custom settings - note that the camera has 2 levels of memory banks set up - the 3 'MR' positions plus 4 more 'M' positions. The 'M' positions are all stored on the card, and are not as easy to access, but the 3 MR banks are stored on the camera...stick to using those if you switch cards. And there's one easy way to set up access to the MR setting without having to menu dive - you can assign one of your MR banks to a custom recall button. If you want a still bird/BIF switchability, you can set up the camera to operate in 'still' mode, and assign one of your full MR banks set up for BIF to a custom button. When needing to quickly shoot a BIF, just press and hold that button, and immediately all of your settings are overridden and you're in BIF mode - release the button and you're back to regular still bird mode.
My understanding was that you actually CAN store AF mode settings to the MR banks for custom recall - if you set the camera to AF-C on the dial when storing the MR settings, and then when assigning the custom hold recall button, you can turn the switch to AF-S again, and when pressing that custom recall hold button, it will switch to AF-C. I don't have the camera in hand, but it should work that way from what others have told me.
Hope that helps.