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そんなワケねーだろ、このめでてぇ日に
The person who said this line is known for using katakana instead of Hiragana or Kanji. I've found some other conflicting definitions for ウケ but none seem to fit the context of the above sentence. On a side note I assume めでてぇ is a casual form of omedetou?
The "dictionary" form of:
「そんなワケねーだろ、このめでてぇ日{ひ}に。」
would be:
「そんなわけないだろう、このめでたい日に。」
The original phrase is in the Kanto tough guy speech. Occuring twice in that short phrase is the "famous" あい-to-えぇ/えー switch. 「ない」 became 「ねー」 and 「めでたい」 became 「めでてぇ」. This phenomenon has been discussed many times here on SE, so I will just leave it at that.
「ワケ/わけ/訳{わけ}」 here is somewhat difficult to translate all by itself, but if I had to, I would go with "reason" or "ground".
I would, instead, suggest that you remember 「そんなわけ（は/が）ない」 as a set phrase meaning "said no one ever" if I could go somewhat slangish. To use a more traditional TL, it would be "there is no way (that is true/possible)"
This phrase is used very heavily among native speakers. You will often hear the more slangy form 「なわけねーだろ」 as 「そんな」 is often contracted to just 「な」 in very informal speech.
On a side note I assume めでてぇ is a casual form of omedetou?
No, it is not. As I stated above, it is the same as 「めでたい」.
So, the phrase in question means:
"There's no way that's possible on a happy day like this!"
Provide more context if this does not sound right.
"そんなウケねだろ"
="そんな + ウケ + ね + だろ"
="そんな（のは） + ウケ + ねぇ（not） + だろ"
="そんな（のは）(such story) + ウケ(appeal) + ねぇ（not） + だろ(perhaps)"
This "ね" in this context is the abbliviabion of "ねぇ"="ねえ"="ない"
"ウケね" or "ウケねぇ" is a spoken rude language.
The following use of "ね" is entirely different from the use of the above example.
"今朝は寒いね" It's extermly cold this morning.
"ね" here is used for other's consent.
"今朝は寒いね"
"本当！今朝は冷えるね"
The reason why "ウケ" is in Katakana may be the speaker avoid the sequense of Hiragana like "そんなうけねだろ" to make the word "うけ（る）" clear in the written sentence for the reader. "そんなうけねだろう" is also acceptable.
Generally the word "そんな" could be "そんなには(それほどには) less than expected" or "そんなのは (being foreign to something)." The latter is the case in the context of your example sentence, "being foreign to this happy day."
