I figured out the issue.
My .cproject file did not have the "package" field. It was also missing "boardID", "core", and "coreID", so I added those to match was I was using (e.g. "MK22FN512VMP12").
<storageModule moduleId="com.nxp.mcuxpresso.core.datamodels">
<sdkName>SDK_2.x_FRDM-K22F-OM13588</sdkName>
<sdkExample>frdmk22f_dev_composite_hid_mouse_hid_keyboard_lite_bm</sdkExample>
<sdkVersion>2.6.0</sdkVersion>
<sdkComponents>middleware.usb.device_controller_khci.MK22F51212;middleware.usb.device.MK22F51212;platform.drivers.smc.MK22F51212;device.MK22F51212_CMSIS.MK22F51212;platform.Include_common.MK22F51212;platform.Include_core_cm4.MK22F51212;platform.drivers.clock.MK22F51212;platform.drivers.common.MK22F51212;device.MK22F51212_startup.MK22F51212;platform.drivers.lpuart.MK22F51212;platform.drivers.uart.MK22F51212;platform.drivers.gpio.MK22F51212;platform.utilities.assert.MK22F51212;component.lists.MK22F51212;component.serial_manager.MK22F51212;utility.debug_console.MK22F51212;component.uart_adapter.MK22F51212;component.serial_manager_uart.MK22F51212;platform.drivers.port.MK22F51212;middleware.baremetal.MK22F51212;frdmk22f_dev_composite_hid_mouse_hid_keyboard_lite_bm;</sdkComponents>
<boardId>frdmk22f</boardId>
<package>MK22FN512VLH12</package>
<core>cm4</core>
<coreId>core0_MK22FN512xxx12</coreId>
</storageModule>
It is a bug that these are not added to the .cproject file by the new IDE when upgrading the project files. They are present in the SDK example projects.