Interestingly, the name that China and Taiwan compromised on, for the Olympics at least, is "Chinese Taipei". In that context, China rejected the Republic of China (henceforth ROC) name, which Taiwan wanted to keep on using.
I don't know if that mini-settlement on the Olympic name for Taiwan has any implications for the naming of Taiwan in other contexts.
Another answer states that China wants Taiwan to keep using the ROC name, but offers no source in support... China also does object to the island just calling itself "Taiwan" as well though... but apparently in narrow official contexts... like the Olympics. Xinhua (the Chinese state-run news agency) routinely refers to the island just as "Taiwan", at least in their English news.
And of course, as I first commented, China does claim the territory of the island, so even if Taiwan called itself "Chinese Taipei" in every (official) context, China would still not be satisfied without having some actual clout in Taiwan. What China wants/proposes is a system of government for Taiwan modelled after Hong Kong, i.e. "one country, two systems", which is something that Taiwan rejects.
Finally, the question in the last line of the OP's post is actually
Has the People's Republic of China ever given any indication that their longtime dispute with Taiwan would be solved if Taiwan changed its official name and its constitution?
(Emphasis mine.) Well, it depends on the constitution. If it were along the lines that China wants (as mentioned above)... yes, otherwise no. (Hong Kong also has a constitution.) What China absolutely rejects and has threatened military action in case it happens is a declaration of independence of Taiwan. So if Taiwan's putative (new) constitution amounted to a declaration of independence (instead of accepting Beijing's rule)... it would have the opposite effect, i.e. it would escalate the conflict.