The ideal solution would be to invent a bike storage rack that fits in the under storage lockers of all coaches. It needs to be easily removable so can be taken out of the coach when not on RRB work and foldable so doesn't take up lots of room when stored out of use at the coach depot. It needs to be very cheap for coach companies to buy, or funded centrally for them by TOCs of DfT. I think everyone would agree that is an excellent idea. All that is probably, relatively, easy to do. The hard part is getting someone to sign the cheque. And therein lies the issue. And that's why the rule is "Bikes can not be conveyed on Rail Replacement services" (paraphrased)
That is not going to happen, things that appear easy for a quick internet post are rarely that way to actually put into practice in any useful way.
Lockers on coaches are not a standard size, nor are the locker doors. None are big enough to fit an adult bike upright*, so at best it would be a cage to put bikes in lying flat.
Any cage would also need some sort of fixed attachment and levering mechanism, or are you expecting a coach driver to lift a fully loaded cage with multiple bikes from the pavement to the coach?
Most of the coaches I drove were fitted with large powered doors, one each side, it was easy to put in a bike laying flat lengthways but any subsequent bikes could only be carried if placed on top of the first one. As the luggage loads I carried were usually moderate I would normally only use the nearside lockers so the offside was available for the infrequent occasions I had people with bikes. It worked OK but I didn't drive any routes with large bike demand, colleagues who did drive a route that would get peaks of people wanting to take bikes (without any sort of pre-booking) had regular issues with lack of space and bikes being stacked on top of each other.
From observation, coaches from private hire companies mostly have manual doors which are smaller with several per side making it much more difficult to fit in any bikes, depending on the design of the coach bodies.
*Overdeckers may just be able to fit smaller adult bikes upright but there are very few of these in the country.
All that only applies to traditional full-size high-floor coaches, smaller coaches or ones with mid-mounted wheelchair lifts have even less space for bikes.