You can fly via IST on TK on those dates (for something over £1000). And it is bookable on the TK website.

Go to a brick-and-mortar travel agent and have them use segment-select ticketing to get both tickets on a single PNR.

TK is £698 on those dates (out in V and back in Q). Good deal and the OP can also earn * miles.

I saw those fares on ITA but it was over £1000 when I tried to book. Where did you see it bookable for £698?

Thank you everyone for looking into this for me.

I think my conclusion with regard to the initial question is that you can't do it (I've tried two physical travel agents and Lufthansa as well as lots of online ones). It may be possible on other days, but not when I'm trying to fly which is a shame.

I'm also exceedingly grateful to people for searching for other Star Allaince flight combinations and providing ideas. However, I have no allegiance to Star Allience (I posted in this forum because the airlines in question are both part of the Star Alliance and I thought I might be missing something.)

I think the best single ticket option available is with KLM (including Air Korea out) leaving at 00:05 on 6/1 for £880. This is cheaper and quicker than Turkish Airlines, and whilst KLM isn't great, Air Korea is quite nice. I therefore have a choice between this, a more risky multi-ticket option and changing my dates.