While the New Zealander has impressed greatly during his stand-in performances since the Dutch Grand Prix as a replacement for Daniel Ricciardo, it is looking unlikely that it has been enough to secure him a full-time F1 drive.
With Red Bull already eager to hand Ricciardo a seat at AlphaTauri next season, Lawson's only real hope of getting a 2024 drive was for the squad to choose him over incumbent Yuki Tsunoda.
But with it understood that Tsunoda and AlphaTauri have agreed terms to continue into 2024, and an announcement expected to be made at this weekend's Japanese GP, it means Lawson will have to wait for that opportunity to come.
Options elsewhere on the grid look remote, with only a possible Williams seat still available if it does not keep Logan Sargeant.
Instead, the most likely route is for Lawson to take on a test and reserve role at Red Bull as he bides his time until 2025 when there could be a shuffle of the energy drinks company's drivers.
Speaking at the Japanese Grand Prix on Friday, Red Bull team boss Christian Horner suggested it was unlikely that Lawson would be loaned out, but indicated there were plans for him to remain working with Red Bull.
"He's in the Red Bull family," he said. "I think there's only one seat left available, and we've done that in the past when, for example, Carlos [Sainz] went on loan to the previous Renault team.
"But I think if he's not sitting in a grand prix car next year, then he is going to have plenty on his plate with other stuff to do."
Horner was full of praise for the job that Lawson had done at AlphaTauri, having been thrown in the deep end at Zandvoort after Ricciardo broke his hand in second practice.
"I think he's done an outstanding job," said Horner. "Dropped in in the rain in Zandvoort: that was a bit of an eye-opener for him. But I thought he did well there.
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"Then, a week later in Monza, he was unlucky to miss out on a point. And then seeing Singapore for the first time and scoring a couple of points. He's a gritty racer. We know that about him.
"I think he's grabbed this opportunity in F1, and it's very seldom drivers get a chance to demonstrate their talent. I think he's done a tremendous job and certainly has put himself firmly on the radar, and cemented the feeling that we had about him."
While Lawson looks set to miss out on a race seat this time around, Horner said there was no doubt he would be in F1 in the future.
"We've got a luxury problem where we've got three into two in that team," he said. "Obviously it is an AlphaTauri issue, but there's three great drivers and it's a nice headache to have.
"He will be a grand prix driver, and he already is a grand prix driver, at some stage.
"Whether he has to wait a little for that or not, then I think he's demonstrating that he is a talent for the future."
