Just a few other options for your consideration...

You might also consider taking the train from FRA to Berlin. There are a number of trains daily which run directly from the FRA airport station to central Berlin, and they take around 4 hours. If you take the deep discount fare ("Super Sparpreis"), it will cost as little as €60 in 2nd class and €70 in 1st class. However, these cheap fares are even more inflexible than the LH fares: if you miss the train, you can eat the ticket for lunch.

I also checked the rail connection from Budapest to FRA. The shortest is indeed 10:15. However, the price is very reasonable: as little as €60 in 2nd class and €100 in 1st class. If you like long stretches of looking out the window and seeing the scenery go past, it might be OK. These connections generally involve changing trains in Vienna, and you do have the opportunity to schedule you trip to give you a few hours there, in case that's interesting for you.