たべるだけ doesn't mean "eat as much as possible", it means "just eat / only eat". Ex.) りょうりができあがりました。あとはたべるだけです。 (The food is prepared, the only thing is just to eat.) "eat as much as possible" = できるだけたべる *Though I wonder if my English is correct or not, I hope you can understand this.

" as much as possible " this usually is used at like the following situation, There is much spaghetti in a dish. I can't eat all this much. But I'll eat as much as possible. わたしはこんなにたくさんだべられません。　でもたべられるだけたべます。 - additional notes - たべられるだけ　but usually young people say recently　 たべれるだけ.　（without " ら"） -　just eat - "Let's go have have a drink after work" " I can't drink, just eat is OK?" ”しごとのあとのみにいこう” "わたしのめないの。　たべるだてでいい？"

Good question. I too am eagerly awaiting a Japanese speaker to answer.