一家吃川菜的饭馆“ -- attention to 吃川菜的, not other type of food 吃川菜的一家饭店” -- attention to 饭店, not 饭铺 or 酒楼

“一家吃川菜的饭馆”一般中国语言不会这么说，直接是一家川菜饭馆，是指它是一家做川菜的饭馆；而“吃川菜的一家饭店”是指你在之家饭店吃川菜，这家饭店不仅有川菜，还有别的菜系。

we say "一家提供川菜的饭馆"，or “提供川菜的一家饭店/馆”，but the first one would be better. "饭店/馆" can't eat ”川菜“，but provide ”川菜“。 people can eat "川菜", for example, "到这家饭店可以吃到川菜"

The best answer is ‘一家川菜饭店’. Firstly, for your first sentence '一家吃川菜的饭馆‘, ’饭店‘ is more widely used than '饭馆’. '饭馆' is always used in Chinese history but if you say that we still can understand what you mean but you don not need to add 吃 in your sentence because "一家川菜饭店‘ means ' A Sichuan food restaurant where you can eat Sichuan food'. And for you second sentence, the sentence structure is wrong.

我们很少把“吃”字加在上面的两种说法中。你只要说“一家川菜馆”就可以，大家都就会明白你说的意思是“一家只供应川菜的饭馆，别的菜品这个菜馆不做”。“吃”这个动词是“施动者”发出的的动作，跟“川菜馆”无关。你可以说“这是做川菜的一家饭馆/饭店，”这句话的意思与“这是一家川菜馆”一样。