Everything that my local hardware shop sells is quite large diameter and not very flexible.
Generally speaking, the greater the diameter of the coax cable, the lower the signal loss.
The higher the frequency of the signal, the higher the loss in the cable. From memory, the loss is proportional to the square root of the frequency, eg 4x the frequency gives 2x the loss.
why shouldn't I use oscilloscope coaxial cable which is smaller diameter
What is "oscilloscope coaxial cable" ?
If you mean the cable used for oscilloscope probes, that is not a good cable to use as a transmission line. The centre conductor is a VERY thin wire in order to give a very low capacitance cable.
It may be possible to use a smaller diameter coax cable for your TV.
The things you need to consider are:
The received signal strength
The loss per metre of the cable
The length of the cable
The signal level required at the receiver.
Whether this will work in your case is impossible to say with the imformation available.
One thing which has not been considered so far is the characteristic impadance of the cable.
Television antenna systems are designed around an impedance of 75 Ohm, so you should use 75 Ohm cable.
Another common impedance is 50 Ohm. Using a 50 Ohm cable in your application may be OK, but there is a chance of some rather odd effects happening to reduce the received signal strength.
JimB