Did the Democrat takeover of the Hawaiian government help persuade the Democrats in Congress that it would be OK for Hawaii to become a state?
Not really. I don’t think anyone knew in the mid to late ‘50s that Hawaii was on track to being a solid one-party Dem state. While it had gone Democrat in 1954 for the legislature, the GOP still reelected their House Delegate (the widow of its former occupant), though she lost it in 1956.
At the time of the admission of Alaska and Hawaii, it was Alaska that was considered a Democrat bastion (it had not elected a GOP Delegate since 1930). Hawaii was still competitive for the GOP with its admission in 1959 that it elected the Haole GOP Governor Eisenhower had appointed (Bill Quinn), elected a Chinese GOP Senator (Hiram Fong) and almost elected a Japanese GOP Senator (Wilfred Tsukiyama), the latter losing by less than 3% to an elderly former Haole Territorial Governor (Oren Long, a native of Kansas). Inouye ran against a Haole for the sole House seat and won in a landslide (he was essentially coronated for both offices, both in 1959 and 1962 when Long stepped down).
Conversely, Alaska would take until 1966 to elect a House member and Governor and not until one of its Democrat Senators died in late 1968 did a Republican (Ted Stevens) get appointed (and for the other Senate seat, not until 1980 would a Republican win, Frank Murkowski). So again, the betting back in 1959 is that Hawaii would be a GOP state and Alaska a Democrat one.