You're correct, and those other options are also good. We have the option of stating an indirect object, but that does not relieve us of the responsibility to state the subject (which may or may not refer to the same person) in the subordinate clause. The thing is that the verb "to suggest" is one of very few that are still followed by the subjunctive, while the subjunctive is tending to disappear from English usage, thus creating various workarounds.

I think there's a slight misreading here? "Suggest" requires a direct object, and doesn't take an indirect object. "Suggest me" is incorrect, unless you are the actual idea that is suggested. This is also why we don't use "to [verb]" after "suggest". We need a noun, a gerund or a full clause (and as Phil has rightly pointed out, the clause normally uses the subjunctive verb). When we do use verbs which take indirect and direct objects, we can use the "verb - indirect object - direct object" pattern, eg. "tell us a story" or "buy her a present". If we choose to put the indirect object after the direct object, then we use a preposition (to/for): "tell a story to us" or "buy a present for her". It is perfectly OK to use "suggested to me (that...)", because it is clear that "me" is the indirect object, thanks to the preposition. However, with "me" as the indirect object, we cannot use "suggested me (that we go for a walk)". That is unclear and awkward.