It is without doubt the best idea if (as SJS suggest) you convert the brakes to a nutted fit using the long tektro bolt mentioned above.
But before you do anything else, do check that the new brakes have enough reach to work on your bike anyway. IIRC shimano make lots of brakes but only two dual pivot calipers even have a reach of 57mm, being BR-R650 and BR-R450. All the other brakes they make have a maximum reach of 52mm, I think.
Both types of brake would be rather shorter than would usually fit on a Raleigh Mixte frame; BITD the typical setup would be to have a 610 type caliper at the front and a 750 type caliper at the rear; both have more reach than 57mm. You might find that a 57mm reach caliper is OK at the front but not at the rear on your frame.
IIRC there is a longer reach Tektro DP caliper made in nutted fit which may be suitable. Spa cycles sell these, amongst others. If the reach isn't quite right anyway, it is a no-brainer just to get those TBH.
BTW brakes that are 'just long enough' are a dumb idea if you want good power. In any given brake the mechanical advantage (MA) varies from very good (with the brake blocks in the top of the slot) to nothing like as good, if the brake blocks are in the bottom of the slot. The MA reduces by about 30% in most DP calipers, which is a huge amount. By contrast if you start with longer reach brakes, they have longer arms with more leverage so the MA at (say) 57mm reach is more with a 65mm drop brake than it is with a 57mm drop brake.
Note also that the brake levers need to be the right ones too. Nearly all shimano STI models launched post 2008 (which have 'New Super SLR' cable pull) have a much reduced lever MA and only work properly with matched calipers. Shimano don't make perfectly matched 57mm drop calipers; the two models they do make are not perfectly matched to modern STIs and really were designed to work with the previous generation of brake levers (~1992 to 2007 launch dates).
So if you use BR-R450/R650 at full reach with modern (post 2008) STIs, you can expect a relatively poor brake, by modern standards.
cheers