cant update MacBook Pro (13-inch, 2019, Two Thunderbolt 3 ports) to 10.15.3 (or 2) get error message;An error occurred installing macOS. You can try again to complete the installation;

Your profile shows you have a Mac Pro, but your post says a MacBook Pro. I assume this is correct, so you should change your profile.

Start with:

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the side list. 
4. Click on the First Aid button in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

Download and Install the macOS Catalina 10.15.3 Combo Update. This will update 10.15.1 to 10.15.3.

I have the same problem here with my new MacBook Pro (13-inch, 2019, Two Thunderbolt 3 ports). I have done your suggestion, repair disk from macOS utility, downloaded the 10.15.3 combo update from the apple support, tried to install the combo update both in normal boot and safe boot still failed. I've even tried to reinstall macOS from the utility and it did failed.

What I've done so far:

1. Updated through normal software update from system preferences - failed (the progress bar always stopped halfway, and then it started all over again like I never downloaded it at all)
2. Updated through safe boot from system preferences - failed (same as number 1)
3. Did the first aid to the Macintosh HD in the Utility Menu
4. Downloaded the 10.15.3 combo update you suggested, tried to install it, error halfway.
5. Tried again number 1 and 2 - failed (still the same, stopped halfway and started it all over again)
6. Tried to reinstall macOS from Utility Menu - failed twice

My MacBook was always connected to AC power (if this info necessary). I just bought it 3 days ago. Actually everything works fine, besides this macOS update problem. But I need to update to the latest macOS update so I can transfer my data from my older MacBook Air (mid-2012) which is on macOS 10.15.3. Help please...