Many thanks to fijea at Skyrim Nexus for this guide.
Here's a step-by-step guide for updating Better Vampires with DSR and ASIS installed:

MCM: Mod Configuration Menu (SkyUI)
NMM: Nexus Mod Manager


*Always save on a new slot*

1. [MCM] Dual Sheath Redux > Start/Stop > Uninstall
2. [MCM] Better Vampires > Troubleshooting and Cleanup > Reset/Clear Better Vampires Mod
3. [MCM] Better Vampires > Reset Sanguinare Vampiris
4. Save, exit skyrim.

5. [NMM] Uninstall DSR
6. [NMM] Uninstall BV
7. [NMM] Uninstall Race Compatibility
8. [NMM] Update BV and RC to the latest version
9. [NMM] Reinstall RC without BV (overwrite).
10. [NMM] Untick from load order:
- Dual Sheath Redux Patch.esp
- ASIS-Dependency.esp
- ASIS.esp
(This will prevent ctd after bethesda logo)

11. Load skyrim.
Select "Yes" on missing contents.
12. Save on a new slot (clean save) then exit skyrim.

13. [NMM] Uninstall RC
14. [NMM] Reinstall BV (overwrite)
15. [NMM] Reinstall RC with BV (overwrite)
16. [NMM] Reinstall DSR
17. Patch DSR using SkyProc
18. Reopen NMM
19. [NMM] Make sure ASIS and DSR esps are ticked, including dependency and patch.

20. [NMM] Arrange load order:

Skyrim.esm
---------------
RaceCompatibility.esm
Unofficial Patches
---------------
RaceCompatibilityUSKPOverride.esp
CustomRace.esp
---------------
Better Vampires.esp
Dual Sheath Redux.esp
Dual Sheath Redux Patch.esp
ASIS-Depedency.esp
ASIS.esp

21. Load skyrim
22. [MCM] Better Vampires > Reset Sanguinare Vampiris
(You might need to wait until the MCM menu(s) registered)
23. Wait for 3 days, done!

*You also need to re-configure Better Vampires