Ideally, you would avoid sharing the temporary files (.aux, .bbl, .bcf, ...) with your colleague. Only share the non-temporary important files (.tex, .bib, images, ...). See Matching biblatex in two machines for more explanations why sharing temporary files is usually bad and can go wrong.
Sophisticated systems like Git have a way to ignore certain files and so does Dropbox https://superuser.com/q/469776. Even if your preferred method does not have a way to selectively ignore a file, a simple script that delete temporary files before submission might be simpler here.
If you must share the temporary files, a simple clean-up script that get rids of temporary files before compilation (see for example Deleting external/auxiliary files?) sounds like the best solution to me. Of course this file would not be called before every compilation, just before the first compilation after a sync. You could make the script cleverer by checking if the versions are wrong first (you'd have to parse the right part of the .bbl and would have to compare with the expected version of your biblatex installation, which seems possible, but fiddly).
Here is a LaTeX-based solution that overwrites .bbl files of the wrong version with an empty file by opening it and closing it immediately afterwards.
\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber]{biblatex}
\addbibresource{biblatex-examples.bib}
\makeatletter
\newwrite\blx@jul@deletefile@out
% copy of the original macro from biblatex.sty
\def\blx@ifsigned#1#2{%
\begingroup
\let\blx@tempa\@firstoftwo
\edef\blx@tempb{\csuse{blx@sig@#2}}%
\edef\blx@tempb{\detokenize\expandafter{\blx@tempb}}%
\openin\blx@bcfin #1.#2\relax
\ifeof\blx@bcfin
\else
\endlinechar\m@ne
\readline\blx@bcfin to \blx@tempc
\ifeof\blx@bcfin
\else
\ifx\blx@tempb\blx@tempc
\readline\blx@bcfin to \blx@tempc
\edef\blx@tempb{\csuse{blx@ver@#2}}%
\edef\blx@tempb{\detokenize\expandafter{\blx@tempb}}%
\ifx\blx@tempb\blx@tempc
\else
\blx@warning@noline{%
File '#1.#2' is wrong format version - expected \blx@bblversion}%
\let\blx@jul@deletefile@do\@empty% <- added
\fi
\else
\blx@error
{File '#1.#2' not created by biblatex}
{This file was apparently not created by biblatex.
Rename it or\MessageBreak move it to a location were
TeX will not find it. If this error\MessageBreak
persists, consider redefining \string\blxauxsuffix.%
See the biblatex\MessageBreak manual for details}%
\let\blx@tempa\@secondoftwo
\fi
\fi
\fi
\closein\blx@bcfin
\ifundef\blx@jul@deletefile@do% <- this conditional is new
{}
{\blx@warning@noline{Deleting file #1.#2}%
\immediate\openout\blx@jul@deletefile@out #1.#2\relax
\immediate\closeout\blx@jul@deletefile@out
\global\undef\blx@jul@deletefile@do}%
\expandafter\endgroup\blx@tempa}
\makeatother
\begin{document}
\cite{sigfridsson}
\printbibliography
\end{document}
I wouldn't do that, but if you insist on a LaTeX solution this is pretty universal.
.bbl, but also.bcf,.aux, ...). Related: https://tex.stackexchange.com/q/410512/35864. If you only share the.texand.bibfiles you (1) have to share and transfer fewer files and (2) should both be able to compile with your local versions with less hassle. – moewe Jul 13 '18 at 11:52