Sorry but I am confused. You had quite a bad off on your bike from what you have written, hitting your head (sans helmet) in the process from which you are now suffering blackouts. Why on earth would you wish to claim for a hire bike when you presumably are NOT in a fit condition to push a bike let alone ride one? I think any third party insurer would see through your claim in a second. Claiming you need a hired bike whilst yours is out of action would certainly weaken any subsequent personal injury claim. Yes you have to mitgate your losses, but claims also have to be truthful and reasonable. The fact that you have another bike makes no difference. If you were fit to ride you could claim a daily sum for loss of use of your Cube bike until it was replaced or repaired. If you are not yet physically fit to ride a bike you could claim for travel expenses such as public transport or even taxis if you are too fragile to travel to and from work in the scrum of the great unwashed, but given your description of your injuries and continuing symptoms I feel sure you are not yet back at work. If you are, you shouldn't be until assessed to be fully fit and given the all clear especially regarding the black outs you now experience which are SERIOUS. You really have to get your priortities right. Have you contacted any PI solicitors? For if you have, they will or should have, explained all this to you. It's all very well saying you worked in non fault accidents, but in what capacity? If you follow your own advice and cock your claim up you are screwed, where as if you instruct a solicitor and they screw up you can sue them for being negligent.
Sorry to appear harsh but having been knocked down myself and spending 4 years claiming against a 3rd party, a Mr Frank Leigh, and obtaining just over £20k, I like to think I know what I am talking about. Your health is by far the most important aspect, DO NOT rush to get back on your bike until absolutely fully recovered and fit. Compensation is a crude means of trying to put you back as you were immediately prior to the collision which was caused by the other party. You only get one bite of the cherry so to speak so you don't want to f*** it up ............. From what you have written if you obtain the correct advice and are properly represented you could be in line for a substantial award of damages and I mean substantial. Unfortunately it will take some time especially if you now suffer black outs as your prognosis might be uncertain. It might mean that you will have to surrender your driving license or you are not able to work in certain employment where a risk of blacking out could be seen as too greater risk such as driving or operating machinery. If this is the case then to be disadvantaged in the jobs market and as a result suffer a reduction in future earnings due to the other party's careless and negligent driving, could mean, you are be entitled to compensation for this which might be substantial.
You need specialist advice from a good PI solicitor preferably a cycling one. You have been given sufficient names IIRC.
Anyway get well and DON'T go riding your bike until you are fully recovered. And buy a helmet for when you do return to cycling ................... although horses and stable doors now come to mind.