Is there an alternative way to perform the same function as UpdateApplication on Android 15?

I don't think you realistically are going to be able to continue to offer silently self-updating mobile applications in the future. These updates are expected to be performed by an EMM tool and/or Managed Play moving forward. I do not think declaring your own (non EMM agent) application as DO or PO is an appropriate resolution to this problem.

DO under AE operates very differently from DA with one of the core differentiators being their can be only one Device Owner on a device. You would not be able to rely on declaring such a permission given that many of your end clients devices will likely already be managed by a true EMM DPC with declared DO privilege. In addition, DO is something that has to be declared while a device is in a factory default state. You cannot install your app later and then declare DO permission as it would otherwise already be too late. 

Inflating your own Work Profile just for your own app and then declaring that app the PO of that Profile also doesn't seem to be a valid solution in my mind. I could be mistaken about that approach however since I'm less familiar with PO deployments.