I'm sorry to say but there is no word like 猫舌{ねこじた} for spice in Japanese language.
I think words are born and exist to distinguish special ones from ordinary.
In conclusion, the person with 猫舌 is special, but the person who is sensitive to spicy food is not special in Japan.
We call a person to be 猫舌 who cannot eat or drink somewhat high temperature things that many/ordinary Japanese can eat or drink.
On the other hand, observing the sensitivity to spicyness, many Japanese people cannot eat very spicy food. Therefore, for majority Japanese, we don't have interest in those who are sensitive to spicy food, so I think that the word itself does not exist.
When we want to express that I cannot eat spicy food we say "私は辛{から}いのは苦手{にがて}です or 辛いのは嫌{きら}いです", but "甘口です", "甘党です" or "お子様舌です" can not be substituted.
As for 甘口 and 甘党, 甘口 refers to the taste of food or drink including 酒{さけ} and 甘党 refers to a person.
EDIT
There is a topic on food temperature and spicy food, so I'll write what I understand about them.
Examining the English dictionary, "hot" has at least the following two meanings: they are "having a certain degree of heat esp. a high degree" and "causing a burning taste in the mouth". These two adjectives correspond to "熱{あつ}い/暑{あつ}い" and "辛{から}い" in Japanese. The pronunciation is the same as "あつい", but "熱い" means that things containing food are hot, on the other hand, "暑い" means hot weather.
It would be a matter of course for people who understand sensibly the two meanings of "hot", so I think they don't have much doubt about what is written in one of the comments as "辛{から}い物{もの}が苦手{にがて}な人は、熱{あつ}い物も苦手ですか? Cannot the people who cannot eat spicy food eat hot food?"
Actually this comment is talking about very interesting things that Japanese people don't know so much.
The reason is that many Japanese who are not familiar with English understand these two adjectives or feelings as completely different things. Also, many Japanese don't know that these two adjectives are the same word with "hot" in English.
Even Japanese who know that the two Japanese adjectives are the same word in English as knowledge should not understand sensuously as the same thing. They know they would sweat as soon as they eat very spicy "カレーライス curry and rice" or "ラーメン ramen", but they would not think that the cause of these two adjectives is the result of stimulating the same sense organ is. Actually it may be scientifically true, but I think many Japanese people, including me, will find it interesting to know the above comment.