In a nutshell it seems like a simple question, If a person detained under section 136 at a Place of Safety, awaiting assessment commits an offence should they remain, await and complete the assessment or go to custody? The circumstances would not be such that the person was a serious risk to themselves or others or that it was not a suitable place for them to be detained due to the levels of their behaviour.
THere is no single answer to this - it will depend on the specifics of the person, the offence and the background risk factors. As a general point, the more unwell the person seems to be and the less serious the offence or background, the less the imperative to instantly remove to custody for investigation. The more serious the offence and / or the risk factors, the greater the consideration should be for custody.
Nothing prevents any person being investigated for the offence, regardless of whether you remove them to custody immediately; and nothing prevents a patient who is not further detained MHA from being arrested, if the grounds for doing so still exist at the point where the s136 ends.
All situations turn on their individual merits and the above summary takes account of public policy on the diversion of mentally disordered offenders (awful terminology).