I'm not a K1 visa holder, but I did recently get a learner's permit from the Louisiana OMV. If your existing DL is from the UK, you will likely need a learner's permit as well. Check with the OMV staff if you will be allowed to take the road test with your UK DL without getting a learner's permit and going to driving school first.
You cannot get a Louisiana DL/learner's permit/state ID as a K1 visa holder (prior to I-485 filing). On Table 2 of the PDF you linked, for K1 -- "No Issuance by OMV". Instead, you should apply as an "Applicant for Permanent Residency I-485" (listed in Table 1). Don't even mention K1 as it will just confuse the OMV staff.
As an I-485 applicant, bring the following original documents to the OMV --
Foreign passport
SSN card
Unexpired EAD card
I-797 (I-485 receipt notice)
Proof of residence (any US government/bank/insurance-issued letter with your name and Louisiana address)
Lastly, make sure you have a confirmed appointment before you go to the OMV. No need to print the appointment confirmation. Just show the staff the OMV email or text message on your phone. At the branch I went to, those with appointments did not need to line up. We sat at the waiting area until our name was called. Those who had no appointments for that day had to wait in line outside until all those who had appointments completed their business at the OMV.