How to manage app compatibility on iPhone SE without updating to iOS 16?

When the iPhone SE (1st generation) came out in March 2016, it ran iOS 9.3.

It got six major iOS updates and can run the latest release of iOS 15. Six major updates is more than most iPhones get, but

• Your phone lacks a Neural Engine. All iPhones that can run iOS 16, 17, or 18 have one.
• Your phone has only 2 GB of RAM. All iPhones that can run iOS 17 or 18 have at least 3 GB of RAM (and the ones with only 3 GB of RAM are stragglers which may have barely made the cut).
• Your phone's processor is much slower than the ones in current iPhones.

Developers decide whether their apps are backwards compatible or not with previous iOS versions. You should contact the developers of the apps that require an iOS update.

(I am assuming you have the first version of the SE which cannot run iOS 16)

It seems you're interested in updating your iPhone to iOS 16. Is there a specific reason you'd prefer iOS 16 over other versions, even though it might not be compatible with your device? Perhaps you're encountering limitations with apps that require a newer iOS version.

Apps that require higher iOS versions than your device can be updated to... - Apple Community

These are the iOS options based on an iPhone Model

• iOS 18.3.1 —> iPhone models compatible with iOS 18
• iOS 16.7.10 —> iOS 16 and iPadOS 16 are compatible with these devices
  • Chipset A11 - 8 (2GB RAM) / 8 Plus (3GB RAM) / X (3GB RAM)
• iOS 15.8.3 —> iOS 15 is compatible with these devices.
  • Chipset A9 - 6S (2GB RAM) / 6S Plus (2GB RAM) / SE (1st Gen) (2GB RAM)
  • Chipset A10 - 7 (2GB RAM) / 7 Plus (3GB RAM)
• iOS 12.5.7 —>
  • Chipset A7 - iPhone 5S (1GB RAM)
  • Chipset A8 - iPhone 6/6 Plus (1GB RAM)
• iOS 10.3.4 —>
  • Chipset A6 - iPhone 5 (1GB RAM)
• iOS 10.3.3 —>
  • Chipset A6 - iPhone 5C (1GB RAM)
• iOS 9.3.6 —> iPhone 4S (512MB RAM) - Chipset A5
• iOS 7.1.2 —> iPhone 4 (512MB RAM)- Chipset A4

Which means the "old" version that was compatible with the first SE is no longer available.

So you would need to contact the developer.