A would opt for a totally enclosed motor, a TEFC the cooling probably becomes an issue below around 15 Hz, a TENV would be more optimal if going below 15 Hz. A TEBC may be more ideal over a wide RPM operating range, but much more expensive and larger. Most newer inverter motors have a something like 10:1 constant torque so that would not be an issue (vector motors are 1000:1 or higher), but at 300 RPM you are only getting around 0.8 Hp from a 5 Hp motor. Most 4P inverter motors in the 3-5Hp range will spin up to 2X their base speed and vector motors to 3X their base speed and maintain their Hp. So you would be better off changing the pulley ratio of the wood lathe to over speed the motor to something like 2 X its base speed (or 3X with a vector motor). If you are using a 3600 RPM motor currently, then go with an 1800 RPM motor and over speed it. Typically at speed you will be turning smaller diameter work and torque is not an issue, low speed larger diameter and torque/Hp is an issue. I would look at 2 options based on the pulley size ratios you can fit.
First would be to use a 3 Hp vector motor and use it from something like 30-180 Hz. You will have 1.5 Hp and full torque at 30 Hz, full Hp at 60-180 Hz. The cost of a 3 Hp VFD is about 1/3-1/2 of a 5/7.5 Hp VFD and the motors can be purchased for around 600-700 or less. I purchased the motor below for a lathe (accepted a lower price), also look at the BlackMax Y541 and the Baldor IDNM series. The physical size of the motor will be smaller and shipping costs less.
LINCOLN LM06119 INVERTER DUTY ELECTRIC MOTOR 3HP 1800RPM 3PH 60HZ TENV 184TC | eBay
Second would be a 5Hp inverter or vector motor, so consider something like TENV or TEFC, run the motor from 20-120 Hz. At 20 Hz you will have ~1.7 Hp, but much more torque. There are lots of 5 Hp motors in this category at a price close to the 3 Hp above, but they are larger and heavier. There is the option of going with a rolled steel case TEFC like the Baldor below which would work fine in this application and are less expensive and weight/motor frame is less then say a cast iron TENV motor.
LINCOLN LM06125 INVERTER DUTY MOTOR 5HP , 213TC , 3 PHASE , 1800 RPM , TENV | eBay
EM3615T BALDOR 5HP MOTOR 36G271S268G1
VFD options, as others have mentioned you would be looking at a 3 Hp then something like the WJ200-022SF, Teco A510 or E510 (E510-203-H1FN4S-U is their NEMA 4/4X/12 Series), Fuji Electric, etc. These range between $300-500 based on the enclosure rating for a 3 Hp single phase input. If using a 5 Hp motor, I would go with the 7.5 Hp 3 phase input VFD and derate for single phase. Yaskawa does make a single phase input VFD, CIMR-VUBA0018FAA that I have used, so also an option but you are looking at the $600-700 range either way you go and you need a larger service and associated costs. You will probably want an external braking resistor for the VFD, in particular if you are turning heavy materials.
Yaskawa CIMR-VUBA0018FAA, 5 HP, 200-240V, VFD
Hitachi, WJ200-055LF, 7.5 HP, Variable Frequency Drive 230 Volt, 3 Phase Input, IP20, at Dealers Industrial
With 3 phase input VFDs running on single phase or off of a RPC, using a DC buss choke is recommended to diminish the increased harmonic distortion and also decreases the input fusing requirements due to the current peaks. You are not using the VFD at it's maximum ratings for any length of time, so there is a wider margin of reserve for the VFD in this application so a VFD derating of 1.5 for single phase input is acceptable.