What is the reason for the sudden increase of French published accident and incident investigation reports in 1997?
A related event could be the adoption by The European Community Council of:


Council Directive 94/56/EC of 21 November 1994 establishing the fundamental principles governing the investigation of civil aviation accidents and incidents [Official Journal L 319 of 12.12.1994].


European directives must be transposed in a country law and adopted by the country Parliament. This has been done for France in 1999, with the law "Loi n°99-243 du 29 mars 1999". This law actually modified the regulations related to civil aviation (Code de l'Aviation Civile, Livre VII).

Most of the articles added new obligations to the State and, from what I understand, no obligation for investigation or for releasing a public report existed before, so it may have been at the discretion of the Transport Ministry.

Two articles are of interest:


  Obligation to investigate every accident or serious incident with the
  aim of preventing any reoccurrence thereof.


and 


  Obligation to publish an accident (or incident) report, containing
  safety recommendations where appropriate.




Regarding your count on the BEA site, this info is lacking some important details, for instance:


The hang-glider or ultralight aircraft are relatively new, and we don't know how it has changed the number of incidents/accidents. It's well known that in France hang-glider accidents where not investigated by the BEA because hang-gliders are not subject to an airworthiness certificate. It seems this has changed, but it's not clear to me.
You don't provide a list of events, only a list of reports, which as demonstrated in the Rio-Paris accident can multiply by 4 the count because of the 3 interim reports (this is now mandatory to publish a preliminary report each 12 months if the final report cannot be completed within the first year).