as the pronunciation for L is <el>, I suppose "an LV purse" "an LV handbag" etc are  correct forms

i I think should be a

an LV, as the first sound (phoneme) in the word is /e/, a vowel.

'An' goes before a singular noun in whose pronunciation the first sound is a vowel, but not necessarily a vowel letter. Note the difference between a vowel sound and a vowel letter.

Similarly, an L-shaped rod,
an X-ray photo   (first sound /i/)

On the other hand,
a university (first sound /j/) (though u is a vowel letter).