No, won't make any difference to the firmware at all. They're two separate things:
- Network lock is what is soft coded and ties it to using a specific network provider (by ICCID). Getting an unlock code just removes the restriction and allows other providers SIMs to be accepted - you've already got that as your using EE on a Voda configured handset.
- Firmware Carrier CSC is the physical Firmware on the device. Service providers have their own variants (to apply network specific elements and bloatware) on devices they supply. Handsets direct from Samsung, Currys, Argos etc are all true SIM free units and use a generic open (non service provider) Firmware - this is what is referred to when the term 'unlocked handsets' is used referencing Beta.
The only way of changing your Firmware to be eligible for Beta is by having your handset Firmware flashed (replaced) with an unlocked Firmware version. To be honest, going to that extent just to get on Beta testing for a few weeks is an extreme move. The official release of A13 and OneUI5 will be out in a couple of months and a phased rollout to the 22 and 21 series.