centered text in sticky labels package

I’m trying to print a bunch of labels and using the LaTeX labels.sty style documented at http://tug.ctan.org/macros/latex/contrib/labels/labels.pdf

These are not for street addresses (which are naturally left-justified), but rather to put on chess trophies (I found gold leaf labels).

Unfortunately I cannot figure out how to make the text centered, and left-justified does not work so well for a trophy label. The documentation does not seem to address it.

In the labels.sty source I see that line 219 begins a tabular environment with:

 settoheight{LabTmp}%
  {begin{tabular}{l}usebox{this@label}end{tabular}}%

but if I change the {l} to {c} the results are inconsistent and not really what I want, almost as if only one of the lines was centered.

Does anyone know if that can be done? Or if I can put some special control characters in the label data file to nudge the spacing a bit?

[EDIT: adding MWE; Peter, thanks for pointing that out]

[note: the two solutions offered so far are very good, but merging with my MWE is a problem: I use TopPageMargin and BottomPageMargin to adjust to my printer’s offsets and those macros fail if I use the xpatch approach]

Here is the file labels-trophies.tex:

documentclass[12pt,letterpaper]{article}
usepackage[newdimens]{labels}
LabelInfotrue
TopPageMargin=16mm
BottomPageMargin=12mm
begin{document}
LARGE
LabelRows=10
LabelCols=2
labelfile{trophies-mwe.dat}
end{document}

and here is a small label file trophies-mwe.dat:

NM K-9 Championship
2014 State Champion

NM K-9 Championship
2014 State Co-Champion

NM K-6 Championship
2014 State Champion

NM K-6 Championship
2014 State Co-Champion

NM K-3 Championship
2014 State Champion

NM K-3 Championship
2014 State Co-Champion

From this I get:

markgalassi@mozart:~/h/misc/chess/state-champs/2015$ pdflatex labels-trophies-centered.tex 
This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=pdflatex)
 restricted write18 enabled.
entering extended mode
(./labels-trophies-centered.tex
LaTeX2e <2014/05/01>
Babel <3.9l> and hyphenation patterns for 2 languages loaded.
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))
(/usr/share/texlive/texmf-dist/tex/latex/labels/labels.sty)
! Undefined control sequence.
l.4 TopPageMargin
                  =16mm
? 

documentclass{article}
usepackage{labels}
usepackage{xpatch}
makeatletter
xpatchcmd{start@@label}{raggedright}{centering}{}{}
defstart@newline{par}%
makeatother

begin{document}
begin{labels}
Me
My address
My City, State, Zipcode

My Brother
His address
His City, State, Zipcode

You
Your address
Your City, State, Zipcode
end{labels}

end{document}

documentclass{article}
usepackage{labels}
usepackage{xpatch,ragged2e}
makeatletter
xpatchcmd{start@@label}{raggedright}{Centering}{}{}
%xpatchcmd{end@@label}{end{minipage}}{parend{minipage}}{}{}
makeatother
begin{document}
begin{labels}
Me
My address
My City, State, Zipcode
My Brother
His address
His City, State, Zipcode
end{labels}

end{document}