Well I am afraid that the position is pretty much as the briefing implies.
The earlier Coms decision (Not an HB/CTB one) suggests that you don’t necessarily need a new claim form, but the later HB/CTB one appears (IMO anyway) to confirm that you do.
In the light of these decisions I am afraid that the “safe” option is to require another claim form.
The HB/CTB decision suggests that if you don’t get a new claim form then any payments made after the break in claim is decided are not payments of HB and/or CTB. They would therefore attract no subsidy, and are most probably illegal depending on your LAs standing orders and delegated authority.