But with IV, the 1986 date also seems to come from TV ads and other promotional material, and Leonard Nimoy.
Granted, there's no reason it couldn't be in 1986, and that's the simplest assumption to make, but neither is there any reason that it couldn't happen a couple of years earlier or later.
I found the below on Memory Alpha
According to dialogue, the film begins at some point during the third month of the crew's exile on Vulcan, after the end of Star Trek III. Kirk makes a reference to the 1789 HMS Bounty mutiny, having occurred five hundred years ago. Gillian from 1986 mentions that she has three hundred years of catching up to do. Suggesting that, Star Trek IV takes place in the mid to late 2280s, around 2286 or 2289. Television ads for the film specifically say, "Stardate: 1986" indicating the year, and Leonard Nimoy, in an interview at the film's release on "Good Morning America" in November 1986, specifically mentions that the crew journeys back in time 300 years to "now," again confirming the 1986 date.
and:
The date on the San Francisco Register was Thursday, December 18, 1986. This date is correct for that year. [1] However, a prop for the pawn shop ticket for Kirk's glasses was on display at Star Trek: The Experience (but not seen on screen) was dated 19 August 1986. [2] It can go either way.
If the date for the newspaper was seen on screen, that would likely make it canon--the pawn shop ticket was not seen on screen so if I had to give weight I'd go with the December 18 date.
At one point I thought I saw something that said the 20th century portion was in 1987 but I can't remember where. I checked my novelization quickly but didn't notice anything with a quick review.