The fighter radar isn't a rotating dome, it has coverage limitations, so it cannot be expected to consistently paint the target at all times in an engagement. There are also situations where the fighter's radar has to be shut-off (from risk of passive-seekers, enemy RWR, enemy EW jamming, etc).
In modern, 21st century air combat, such loss of situational awareness would be unacceptable. Whether the radar on the jet needs to be turned off or not, under no circumstance is it acceptable to lose situational awareness. Whether it is through AWACS link, passive sensors, or network centric capabilities, the pilot will continue to have up-to-date information on the bogey until the kill is confirmed. If this cannot be done, then its time to upgrade the EW suite, or get a new plane.
The more you can pack into the munition and increase its effectiveness as an autonomous weapon, the better.
This is a major reason why 5th-gen WVRAAM are a dangerous weapon; HOBS enables the pilot to deploy the missile and then execute a lock (via HMD/S) after launch. Once the target comes into the view of the IIR seeker, the AAM becomes a major problem for the recipient.
The idea that this could potentially be done to a BVRAAM via an AESA seeker with wide and long-range radar coverage of its own is enticing.
BVR and WVR are two separate beasts and it is best not to mix them up. WVR missiles enjoy the luxury of supersonic/hypersonic speeds throughout their flight because of the short distance they need to cover. This, combined with advanced seekers is what makes them lethal. There simply isn't any equivalent in the BVR world. You would never use a BVR missile in a WVR situation, except if you are completely out of missiles.
The range of a BVR depends, amongst other things, on which fighter is wielding it.
http://ausairpower.net/APA-Rus-BVR-AAM.html
In terms of kinematic performance, a key factor which is almost universally ignored by Western planners other than the F-22 and F-111 communities, is the impact of the launch aircraft's kinematics at the point of missile launch. A supersonic Su-35 sitting at Mach 1.5 and 45,000 ft will add of the order of 30 percent more range to an R-27 or R-77 missile. Low performance fighters like the F/A-18E/F and F-35 JSF simply do not have this option in the real world, and the reach of their missiles is wholly determined by the parameters of the propellant load inside the missile casing, and the ability of the midcourse guidance algorithms to extract every bit of range from that stored energy. The result of this is that an AIM-120C/D which might look better on paper compared to an equivalent R-77 subtype will be outranged decisively in actual combat.
It is no wonder then that a single engine jet such as Gripen-NG today boasts supercruise capability. In the future, the range and lethality of BVRs may be increased not by improving the missile, but by improving the platform that launches it. The future of BVR that I have read about foresees a spearhead fighter controlling BVRs launched by other planes - possibly even the AWACS itself. Sans missiles hanging on external hardpoints, the aircraft would have low RCS, yet able to wield the deadly arsenal launched by a different plane.
Yes the pilot will still be in the loop of executing the engagement, but he can switch-off his radar and also prepare for evasion or WVR whilst executing a long-range attack without making the BVR engagement hopeless (which is what happens when the AAM is still relying on the launch fighter's radar for mid-course data).
The bottleneck is seeing through advances in power consumption, which returns to my original point of investing in seeker (and by extension power management) technologies.
The BVR missile need not rely on the fighter's radar. It can rely on an AWACS. Or, it could be following a jamming signal. Or, in the case of later model Russian missiles, it may have an IR seeker.
Let us recall the current mantra of F-22 and F-35 pilots: the Russians like to boast there is no such thing as a stealthy plane, but we don't need to make the plane stealthy. We just need to attack the weakest link on the enemy's kill chain. My personal opinion: In a world of high powered radar jamming, microwave interference, active cancellation, and radar based decoys, the weakest powered component is literally the weakest link.