Yes."would like to" ... Is it OK not to use the phrase?
You can get rid of "that", too:"We(削除) would like to (削除ここまで)suggest that you provide more details."
Does the Department have the power to demand more information, or is it advising that it may be in the driver's interest to provide more information, to support the driver's argument?the department asks for more information
The Department has the power to demand more information, otherwise it is unable to follow up on the incident.Does the Department have the power to demand more information, or is it advising that it may be in the driver's interest to provide more information, to support the driver's argument?
I suppose, the letter should make this clear. The wording will depend on it.
So "we would like to suggest" is not true. You can say "The law requires you to provide further information..." Government agencies have to be crystal-clear about the exact scope of their rights and powers, and the specific legal source of those rights and powers.The Department has the power to demand more information, otherwise it is unable to follow up on the incident.