Where a vehicle is damaged it can fall into one of 4 categories - A, B, C or D.
A - Total loss - vehicle usually burnt out
B - Only fit for breaking for spares - so badly damaged it would be unsafe if repaired
C - Beyond economic repair - BER - new replacements/labour more than vehicle is worth
D - Vehicle less than 1 year old that owner wants replaced
Bearing this in mind, it is perfectly feasible for you to be involved in an accident, have your car deemed BER and still drive it. Lets say that the car is worth €1300 and repairs come to €1500, (as per your post).The quote for repairs to your vehicle is for new parts which you are entitled to - but that doesn't mean that you HAVE to use new parts. The vehicle is still yours and you are entitled to keep it, repair with used parts and still drive it around.
The insurance company are putting you back into the same position as you were prior to the accident by settling you as though the vehicle is gone, less the highest bid for the remains.
You have the logbook etc and are able to tax as usual. You should get the repairs done and then take for an NCT, but are under no obligation to do this, although I would advise for your own piece of mind.