If anyone in the future runs into this issue:
- Install Visual Studio 2019,
- Instal SQL Server Integration Services via the extension menu in VS 2019,
- Open the SSIS project in question, open properties, change deployment target to SQL Server 2022,
- (Possibly optional) Upgrade all packages by right clicking on the ssis folder and chosing the upgrade option.
After this, I have been able to open my project in Visual Studio 2022 Community [preview] without getting any of the errors from before.