Translation

Hello! Could someone tell me how to say this in japanese:

"I'm really sorry for running over your dog"

It would be good if it was not too long, its should fit on a cake

Excuse me for saying this, but you ran (possibly with a car or a bicycle?) over someone's dog, and you are going to present them with a cake with that message on it? Umm, I'm sure it started with a friendly idea, but putting that kind of message that reminds the recipient of a sad and upsetting experience on a cake is not really appetizing, if you know what I mean...

If you must, "wan-chan wo hiite shimatte, gomennasai."

(I'm using the word "wan-chan" your doggie, instead of "inu," plain "dog.")

AK is right--it'll be weird. In Japan, a cake is presented in a happy occasion or to celebrate something. Giving a cake to the dog's owner is not appropriate. Give her/him a card.

I wouldn't be sending cake either.

Follow up suggestion and at the same time, a couple questions:
Would this sound weird to write as an apology for the same thing (resulting in wan-chan's death)?

”この度、ご家族の間、とんでもないご迷惑をお掛けしており、誠に垂ｵ訳ありませんでした。愛犬の成仏を心底から祈ります。”
1. Is it appropriate or not?
2. Does it sound over-apologetic　(too many words, kissing-ass)?

Thanks in advance!

I'd change to:

”この度はとんでもないご迷惑をお掛けしました。誠に垂ｵ訳ありませんでした。愛犬のご冥福を心から祈ります。”

ご家族の間= Among family. It doesn't make sense.

ご冥福 is much better than 成仏.

It's not too apologetic, just polite.