I'm chipping in because I have reasonable adjustments in place at work.
You know your own employer best and if a conversation would be better than a letter but it is important to get any agreement, or lack of one, in writing. If your employer is small and doesn't have procedures already, it may be worth pointing out that a written agreement gives them legal protection because it shows they have met their obligations. It also gives you protection against discrimination.
I can understand reluctance to recognise MH issues as a disability, after all, I wouldn't swap places with someone in a wheel chair but these issues still have a long term negative impact on your life.
Sometimes we are just not up to operating at full steam. That doesn't mean you are not capable of working to a high standard. A bit of understanding by and a few adjustments from, your employer can mean that you can minimise the impact on your work, benefiting both parties.
The important word in these agreements is 'reasonable'. Depending on the job and the employer, this can include things like; more flexible working, opportunities to work from home, an extension to the number of days off sick which would normally trigger HR action, changes to working environment, adjustments to workload, a change in the type of work to minimise anything particularly triggering, a temporary or permanent change of role or reduction in hours worked (with consequent reduction in salary). Phased returns to work are not uncommon either. You can slowly build up your hours and work back to full time levels.