It's not a fixed pattern with an idiomatic meaning, but a combination of simple rules. Taking 食べる as an example verb, 食べはすまい is just 食べはする with まい (negative inference) and か (question marker) attached. 食べていはする is 食べていはする with まい and か.
- 食べはする and so on are ways of attaching は (contrastive), も ("also/even") and すら ("even") to a verb. For example,
食べる + は
becomes 食べはする
("eat at least"), 食べる + も
becomes 食べもする
("also eat"), 食べる + すら + ない
becomes 食べすらしない
("not even eat"). See also: What's the grammatical breakdown of "知りもしないわ"?
- 食べはしまい (or 食べはすまい, 食べはするまい) is a negative inference version of 食べはする. しまい, すまい and するまい are interchangeable. It means 食べはしないだろう. For details, see this question.
- か is a question marker.
Therefore, 食べはしまいか is roughly the same as 食べはしないだろうか or 食べないだろうか ("Won't he eat it?", "I'm afraid that he will eat it").
If there's also a progressive meaning, just replace 食べる to 食べている and follow the same pattern:
- 食べている: is eating
- 食べていはする: is eating (at least)
- 食べていはしまい: it's not likely that he is eating
- 食べていはしまいか: isn't it likely that he is eating?
食べていはしまいか is roughly the same as 食べていはしないだろうか or 食べていないだろうか ("I wonder if he's eating?", "I'm afraid that he may be eating it").