I can not answer your question regarding vim-latex (aka LaTeX-Suite), however, I can recommend two plugins that provide good label and bibliography completion:
Vimtex is a simple and lightweight LaTeX plugin for vim that is initially based on LaTeX-Box. The code is now very much diverged from LaTeX-Box, and the plugin is IMHO better maintained and more actively developed. Disclaimer: vimtex is written and maintained by me.
As for your question, vimtex supports a modular approach where you may enable/disable functionality that you want to use. If you only want the completion functionality, then you may achieve this in vimtex by disabling all the other functionalities. Place the following in your
let g:vimtex_fold_enabled = 0
let g:vimtex_indent_enabled = 0
let g:vimtex_indent_bib_enabled = 0
let g:vimtex_labels_enabled = 0
let g:vimtex_toc_enabled = 0
let g:vimtex_mappings_enabled = 0
let g:vimtex_motion_enabled = 0
let g:vimtex_latexmk_enabled = 0
let g:vimtex_view_enabled = 0
Alternatively, you can disable vimtex completely and then only enable the completion:
let g:vimtex_enabled = 0
In LaTeX-Box one can not selectively enable or disable functionality (I believe), but it is a simple and lightweight plugin with good completion functionality.