Boost Converter unstable with rapid current changes
When working with a boost regulator, the first thing you want to know is the critical inductance and critical current.  Critical current defines the boundary between continuous conduction (CCM) and discontinuous conduction (DCM) in the inductor.  Circuit dynamics are very different between the two modes of operation, and you want to be in one or the other mode.  

Critical current for the boost is approximately:  

\$i_{\text{crit}}\$ = \$\frac{V_o T_s}{16 L_{\text{crit}}}\$  

In this case with L = 1.8uH, \$V_o\$ = 10.5V, \$T_s\$ = 1uSec; \$i_{\text{crit}}\$ would be about 0.37 Amps.  Normally the load current is 0.2 Amps, but pulses to 1.2 Amps.  That's bad.

During the pulse the regulator goes from DCM to CCM adding a pole the the modulator response.  


If the regulator is compensated for DCM, the move to CCM will make it unstable and it will oscillate.  
If the regulator is compensated for CCM, operation in DCM will likely be stable, but the transient response will be very poor.   


To keep the regulator in CCM mode at a current of 0.15 Amps an inductor of about 4.7uH would be needed.  

Another thing to keep in mind is that switching regulators have negative input impedance.  This means that if the impedance of the source voltage is equal or greater than the input impedance of the regulator, the system will oscillate until it runs out of regulation range.  In this case with about 15W of input power from 6.5V, the input source impedance needs to be less than about 1.4 Ohms everywhere below the loop crossover frequency (including any LC resonances).  Looking at the input voltage variations in the pictures, it's not clear that the supply source is up to that.