I take your point on the Anti-Hamas... but that only reinforces why they shouldn't be negotiating a peace treaty, as they are politically and emotively involved.
Qatar's only real benefit is that of distance, in that it doesn't share a border with Israel / Palestine. That and that it does have good ties with both the Arab world and the US (who have a military base there).
Note: I'd quip that Palestinians aren't generally allowed to cross Israel's borders either... but that would be in poor taste.
On the topic of right wing / left wing politics - the Knesset is controlled currently by a coalition: Likud (20), Yesh Atid (19), The Jewish Home (12), Yisrael Beiteinu (11), Hatnuah (6).
Within that (based solely on wiki - so able to be enlightened) I think Likud, Jewish Home and Yesh Atid are all strongly pro-settlement.