Really? Because Nappy III tried to do precisely that between 1865 and 1870 historically, with all his greater executive power, greater political acceptability of militarism, lack of socital/military disruption from regime change, ect. He still coulden't get the resources to do more than start cycling in modern rifles. Add that to the fact that the new Republican regeime is bound to have fewer (and individually less influential) advocates for greater military power and professionalization (As opposed to the "People's army" concept that held the passions of Long Nineteenth Century French Republicans) and the fact that you'll be starting from a lower base number/level of preparedness than the Imperial government and I highly doubt you'd get the even higher expenses of modernizing the artillery, constructing railroads for military purposes, ect.
Furthermore, how belligerent was Prussia acting, really? She'd partaken in fewer wars than France, Britain, or Russia, and the ones that she did (The Schleswig Wars and Austro-Prussian Wars) were internal German affairs in which Prussia's causes had a solid international acceptance and diden't overly upset any regional balance of power. It was Bismark's diplomatic policy doing its job in having kept their metaphorical "Bad Boy" score low, whereas France through her constant meddling and flip-flopping had diplomatically isolated herself. If anything, France looked like a vulture during the mid 1800's, in the sense that she jumped in (In Italy, Mexico, North Africa, over compensation in the Low Counteries during the Austro-Prussian War, ect.) during moments of weakness out of naked greed. A Republican France is only in a better position than the Empire to align with the Italians, and THAT would preclude any continued reproachment with Austria since Rome and Vienna's interests are intrinsically opposed