How to convert a linear dicarboxylate species to the dimethyl ester?
You have a couple of options. 


Your acid-catalyzed Fisher esterification is one of them. The downside is that water, which is liberated in the reaction, has a higher boiling point than methanol. You can't just destill it off to drive the reaction to completion.   
Zhe and  Hexacoordinate-C mentioned diazomethane and methyl iodide which might work well. You might want to make sure that your DNA does not get methylated.
The Steglich esterification, using dicyclohexylcarbodiimide (DCC) and N,N'-dimethylaminopyridine (DMAP) might be an alternative to the Fisher esterifiction.
You might want to try something different, using dimethyl carbonate as a non-cancerogenic alkylation reagent. The procedure is described in Green Chemistry Letters and Reviews, 2013, 61, 89-93 and uses potassium carbonate as a base, dimethyl carbonate as a reagent (and solvent), and tetrabutylammonium chloride as a catalyst.