A mix of brands on the same car might,just might work out ok for a driver who always drives sensibly
in undemanding conditions. But fitting a new tyre, (normally 8mm of tread), on the same axle as a tyre with only 5mm of tread may give unbalanced braking. Even if its the same brand. In heavy rain 5mm might aquaplane or loose grip sooner than 8mm. Same in snow or mud. You could argue that one tyre may be in a deeper puddle than the other anyway or brakes might be slightly uneven. True, but its best not to introduce an additional discrepancy.
Another argument is that premium tyres are better than budget, why would you accept anything less? The reality is not all premium tyres are created equal. Some makes are better in the wet, some better in the dry in snow etc ,or may have other virtues such as noise, tread wear etc. You can never be sure that your tyre is the very best available in every situation. And so a budget tyre may closely match them, and if it performs slightly less well in some respects then at least its predictable and evenly so. And maybe better than some premium tyres. I was not impressed by the pirelli cinturatos I bought a few years ago. That good old Italian tyre maker, now Chinese controlled.
Bear in mind that 5mm of tread the tyre is already half worn out (8mm down to say 2mm is 6mm of usable tread.you have already used 3mm ) It may be better to bite the bullet and replace both .