Between Hainburg and Bratislava I would suggest rather south route because in the north there is detour to
this bridge over Morava/March river also in the south you can better avoid roads shared with cars.
I am not certain if you have plenty of time to visit the city of Bratislava but if you need to find more fluid path then visit the place where the
upper dam on the Danube was erected (interesting modern art gallery there) stay in the south till
Gabčíkovo. Roughly you would follow
this route I did some time ago, probably you would prefer to stay on the side dam all along the Danube where is good asphalt surface and only cars with special permit are allowed.
If you would be interested to visit distributaries of Danube (so called inland Danube delta) then
this route is tested as one where you can pass probably with dry feet. See
this thread for photos and more information. Just in high summer try to avoid it in the very evening or you will be eaten by mosquitos...
During weekend
fantastic food here with a small detour.
https://www.facebook.com/hullamcsarda/ Maybe they are opened whole week during summer but you better check.
Then you would continue on the north shore of Danube, there is marked trail all along to Budapest with only very small part on a road shared with cars. Bratislava - Komárno is described in
this article, there are many photos there, the author had a problem with
gravel roads on his road bike.
Most of it is covered here:
https://www.bikemap.net/en/official/143934-medzinarodna-dunajska-cyklisticka-cesta-slovensko/Budapest – Krakow will ba another post, I need to run now, maybe somebody can suggest something or I will come back here in the evening.