FCL.050 requires flight time to be recorded "in a form and manner established by the competent authority" and, as such AMC1 FCL.050 is guidance and has no legal force. The UK legal requirements for the recording of flight time are contained in Article 79 of the Air Navigation Order 2009.
Article 79 does not require the personal flying log to be maintained in any specific form and it is perfectly acceptable to keep the information electronically and with no hard copy. Neither is there any requirement in law regarding the way that refresher flying with an FI is recorded - it will be sufficient to identify refresher flying as such without the need for a signature.
The only requirement for an FI's signature is in the case of differences training (FCL.710(c)), which is often provided on a separate label for inclusion in a hard copy logbook. It should not be difficult to design some means of including this (e.g. as a scanned copy) in an electronic log.
The above describes the legal requirements - there is no suggestion that merely adhering to these minimum requirements will be either convenient or efficient when dealing with the UK CAA.