I find it hilarious at this point that a WHM would even complain about "losing" Regen mastery to SCH. I do disagree with RDM only being stuck with Regen II at this point. There's nothing "unbalanced" about RDM not being able to get a third tier, especially when they'll still be weaker with it vs. WHM or SCH.
I am genuinely bemused every time I see this. SCH is obviously the best at casting regen spells now, and White Mage's spells are simply not worth casting. White mage has quite clearly lost its mastery over regen to SCH. Aside from the fact that it's yet another set of spells that SCH has quite literally stolen, it represents a massive loss of potential for other regen casting jobs.

Regen deserves a bonus across the board. Red Mage deserves access to at least Regen III, and I would personally say IV, as you say. The spell should scale in duration and potency with enhancing magic. I would further argue that White Mage deserves access to Regen V. The fact that these things are now SCH exclusive means that regen will remain borderline useless for Red Mage and White Mage from now on - wasted potential. SCH has not only stolen the spells, and the mastery - it's stolen the ability to effectively use the spells for the forseeable future. Uniquely. That's not something that should be ignored. If someone can give a good reason for this complete locking out of an entire line of spells then I would honestly love to hear it - nothing that has been said so far justifies this decision by SE aside from laziness on their part.

Regen could be reformed into a fantastic tool with minor adjustments. A simple potency and duration increase, having it not tick down on targets at full HP, and some creative ideas like having an active regen effect "buffer" a fatal strike so that regen can be used as an MP efficient, quick casting, low enmity and above all SAFE healing solution for people not taking damage. But whatever fixes are implemented, they should be universal to ALL jobs who cast regen - otherwise there is simply no point in having the spell in the game. As it stands it simply does not work.

The trouble with all arguments that are in favour of giving SCH unique access (or stolen access) to any spells or tools in the game rely on SCH being a "tactician". Tactical use of magic, specialising in certain fields as required. This isn't as simple as White Mage specialising in restorative/holy magic or Black Mage specialising in destructive/self-serving magic. Using magic tactically means using the right magic, which by extension in SCH's case means being able to cast all magic at a decent potency when needed. There is literally nothing you cannot justify giving to SCH if you go by its professed speciality, as it is - quite literally - a speciality in everything, when you need it.

I've also seen people justify giving SCH mastery over regen based on the job initially learning the spells earlier than White Mage when it was introduced. That just throws back SCH's theft to the moment of its introduction - it doesn't make it acceptable. I'm all for giving SCH unique tools - unique in the sense that they're not potential adjustments to other jobs that SCH has vacuumed up - but to do that it has to have a defined role. Needless to say, "casting other job's spells better than they can" is not really a valid role.