What is the reason for painting ETOPS on the door of the nose gear?
"ETOPS" marking on a B737-800 (source)
Markings on the belly are supposed to be useful for ground handlers, what is it supposed to do here?
You're correct that it is for ground handlers; more importantly, it is for mechanics. It is important to identify the aircraft as ETOPS as the servicing and checking requirements are different for ETOPS and non-ETOPS aircraft. For ETOPS aircraft, two critical systems (like engines) should have differnt maintenance personnel to avoid them making the same mistakes in both of them.
(c) Limitations on dual maintenance.
(1) Except as specified in paragraph (c)(2), the certificate holder may not perform scheduled or unscheduled dual maintenance during the same maintenance visit on the same or a substantially similar ETOPS Significant System listed in the ETOPS maintenance document, if the improper maintenance could result in the failure of an ETOPS Significant System.
(2) In the event dual maintenance as defined in paragraph (c)(1) of this section cannot be avoided, the certificate holder may perform maintenance provided:
(i) The maintenance action on each affected ETOPS Significant System is performed by a different technician, or
(ii) The maintenance action on each affected ETOPS Significant System is performed by the same technician under the direct supervision of a second qualified individual; and
(iii) For either paragraph (c)(2)(i) or (ii) of this section, a qualified individual conducts a ground verification test and any in-flight verification test required under the program developed pursuant to paragraph (d) of this section.
In the same vein, from Boeing:
Strategies for Avoiding Multi-Engine Maintenance Errors
After investigating several multi-engine maintenance events, Boeing identified a key strategy for avoiding multi-engine maintenance error: If possible, operators and repair stations should avoid performing maintenance on multiple engines using the same personnel during a single maintenance visit. Following this strategy should also minimize the potential that improper maintenance will occur on redundant or backup critical systems, such as flight controls, electrical generation and distribution, and hydraulics.
If the NLG doors have ETOPS written on them it would be easier for the technicians to identfy the aircraft and perform checks/maintenance as required, using two different operators, if required.