- You can try to use solver implementing Simplex-method on the base of GMP, https://gmplib.org/, (GNU Multiple Precision Arithmetic Library). E.g. it might be SoPLex, https://soplex.zib.de, see additional info on GMP in SoPlex here https://soplex.zib.de/doc/html/INSTALL.php.
- For a rather "small" LP problem I can you recommend "simplex package" in Maxima (Computer Algebra System based on Lisp). An exact, rational arithmetic is a native for Maxima. See examples and some instructions here https://maxima.sourceforge.io/docs/manual/maxima_311.html, https://maxima.sourceforge.io/docs/manual/maxima_312.html.