Certainly you must travel now. The only question is where.
The places you mentioned will not be worse than Rome. They may or may not be cooler; Budapest, Vienna and Bratislava have pretty similar temperature patterns in the summer--in theory somewhat cooler than Rome. Take a look at the climate-summary charts in Wikipedia, but consider that they almost certainly understate the level of heat you will experience, because those averages are based on data going back to 1991. We are all aware that summers have gotten noticeably hotter since then. The Wikipedia data indicates Krakow is a lot hotter, on average, than the other three cities, but its record highs are lower; I am rather doubtful that the comparison is accurate. As always, I recommend checking out the actual, historical, day-by-day weather statistics for the most recent five years on the website timeanddate.com. That's where you'll find out how many extremely bad days there have been in recent years.
If you can spend some time up in the mountains along the way, that would be smart, and as you no know, getting an early start in the morning is very, very useful.
My trip this year subjected me to an amazing (in a bad way) series of miserably hot days. I started in Albania in mid-May, then I traveled through North Macedonia, Montenegro, Serbia, Hungary, Austria, Slovakia, Switzerland (very briefly) and northern Italy. I was unusually unlucky, but I knew all of those places could be (and some of them definitely would be) extremely hot in the summer. It's just that the heat settled in early and I had virtually no breaks from the heat until about the second week of September. I hope you are more fortunate.