Can an induction motor give out full torque with a VFD?

A good motor with a good variable frequency drive can give 100% of nominal torque at 5% of rated speed, and sometimes at such speed torque should be also higher. If possible, take care to increase cooling not only by convection in air, but also by conduction to cool metal base or metal flange. You can also increase infrared emission using a treated motor surface to increase heat emission (Alumite finishing, or some black paints or treatments). You can also improve the bearing efficiency - if loaded - using some super-lubricants (i.e.: Prolong grease) to reduce friction and related heat generation.

Many of the new motors have a ratio in their data sheets called a turn down ratio which is the ratio between full speed and low speed at full torque. Typical ones are 10:1 at constant torque. If your motor is operating within the turn down ratio all is well. With the presence of the fan you can go even lower provided that the motor manufacture agrees.

The easier one is you will need to overcome the increasing influence of winding resistance (which changes with temperature, of course, as well as with rotor resistance) as the reactance decreases with frequency by a non-linear V/Hz control coming from the FOC algorithm. You may want to consider including an advanced observer or model reference adaptive control scheme to maintain high performance control. The more difficult problem is at slow speeds the rotor heat transfer mechanism is much different than at high speeds due to lack of convective airflow in and around the rotor surfaces. If you are using your independent cooling to move air over the outside of the machine, the rotor will likely overheat at slow speed, full torque. You will have to provide active cooling for the rotor.