Would it be possible for an insect-sized intelligent species to ride a larger insect-sized species that can only survive at around 600°C, and if so, what would the mount need to be kept both warm and safe for the rider?
A (somewhat) equivalent situation is a passenger car pulled by a steam locomotive. The steam locomotive engine is hot, and it has to be. You don't want to ride in the engine, or on it. But you can be pulled along by it. The locomotive can be hot in front and you can stay cool.
Your hot beast will pull a cool insulated carriage. Hopefully you are in hot land and your hot beast is happy going naked as it pulls your smart bug people thru lava land. Your enclosed and insulated carriage might get stuffy but so be it. If you do it right it will stay cool. I am trying to think of something that is not too heavy but which has a high thermal capacity and which you can use to shed heat as it boils away - ah yes; radiator fluid would be fine.
Dismounting will be tricky. You will need a temperature adjustable garage. Your hot beast will pull the carriage in and be detached from it, then it go back outside into the heat to graze the lava grass. The garage then is closed off and the temperature lowered until it is safe for the bug people to emerge from their asbestos carriage.
I'll let aside the practicality of the situation (why the heck would a human need to ride on something that almost melts aluminium?).
The answer is aerogel - over 90% air in a matrix of nanostructured "struts", but the thermal conductivity is a lot lower than the air's, because the micro/nano pores trapping the air don't allow air convection.
Yes, they can be made tough to the point of being machinable with current level of technology
To be able to ride these creatures your intelligent creatures would either need 1-protective refrigerated suits to insulate them from the extreme heat, making them extremely vulnerable to eventual damages to said suit or 2-special suits for their mounts with an internal heater that kept the insides at 600 Celsius, still meaning the creature would be vulnerable should the suit be damaged and unable to keep it heated, plus the risk of damage/breaching on the paws/hooves. I'm assuming here your mount is terrestrial rather than some creature that swims through lakes of extremely hot liquid. If your intelligent species is insect sized, then the mount having a suit scenario becomes much more problematic as small creatures have a natural tendency to loose heat faster, making it more viable to suit up the riders.
Summing up, is it possible? It's very unlikely that we can, with our current technological level, build a special suit that can keep the rider cool without it being bulky to said rider, especially since they're scaled down to insect size. However, if you have the necessary tech to make refrigerated suits capable of keeping the riders cool while also being able to resist the intense heat (as well as some resistance against cutting and perfuration tools) and not reducing too much their natural mobility (keep it at least at the levels of a suit of armor), it should be doable. Though it makes me wonder why your species would choose such troublesome mounts instead of just using their technology to make vehicles for themselves (maybe they need to stay in this hot area and they can't afford to make both heat resistant vehicles and suits?).