There IS a quite flashy McDo in the station. Note that large parts of station were last oct fenced off as it was a near ruin, preparing for a well needed big overhaul. If it rains it does so inside just as well as out.
You may also have noted that a tipical UK hot BF is not that common on the continent. What HUs do is pop into any of the 100s of small supermarkets and buy some rolls and cheeses/ham etc. For your conveneince there is a quite familiair name right behind that McDo: TESCO local! I can really rec. the hot sweet rolls they have-there is always one on special (79/89 huf), tipical HU are the cinnamon buns. I like the turotaski=quark turnovers.
But there are more: a very big SPAR supermrkt is in the basement across and a LIDL further on in a sidestreet behind that. ALL open 6.00/6.30 or 7.00 at the very latest-no daft complaints about unsocial hrs here. Many also have decent cf machines. perhaps even the pizzabytheslice dispenser (199 or was that 299 HUF?) just before the Spar might already be working.
These oso early hrs also apply to about any HTL you may have spent the night before and offering BF. So there is ample time and not even need to hurry much.