I'm mostly familiar with the 'fluid mechanics' part of SL (you probably have seen the names of Lohse and Prosperetti, both I have worked with). So i'm not sure if I can help you a lot, but the way you describe it seems reasonable.

You can eliminate 5, by always aligning with (say) the first big peak (in radius).

I would make a function that has the 5 variables you mentioned as input. Have it calculate a 'cost-function', i.e. the deviation with the model and the data. Then minimize this cost using various methods. Unfortunately the Rayleigh-Plesset is (kinda) slow to integrate, so it might take a while. Maybe try stuff first manually. Then using FindMinimum to optimize.

I don't think it will be easy as you have many many variables. But you can bound radius very well I guess, and also driving pressure you can measure and have a good estimate.

Are you only looking at the scattered light? or also at the produced light? If so, it will become a lot more complex!

I don't have your data so I created my own (similar characteristics):

ClearAll[R]
TMax2=0.00008;
3/2 D[R[t],t]^2+R[t] D[R[t],t,t]==1/\[Rho] ( p0 (R0/R[t])^(3 \[Gamma])-p0-pa Sin[\[Omega] t]+R[t]/cw D[ p0 (R0/R[t])^(3 \[Gamma]),t]) /. {cw->1490,\[Rho]->1000,\[Omega]-> 2 \[Pi] 20 10^3,R0-> 5 10^-6,\[Gamma]-> 7/5,p0-> 101325,pa->13/10 10^5}
sol2=R[t]/.NDSolve[{%,R[0]==5 10^-6,R'[0]==0},R[t],{t,0,TMax2},MaxSteps-> \[Infinity],PrecisionGoal->9,WorkingPrecision->13];
sol2=First[sol2];
Plot[Evaluate[sol2],{t,0,TMax2},PlotRange->{0,All},ImageSize->500,AxesLabel->{"t","R"}]

now we can discretize this data and add some noise, make a copy, and filter it.

data = Table[{t, Evaluate[sol2]}, {t, 0, TMax2, TMax2/10000}];
data[[All, 2]] += RandomReal[2 10^-6 {-1, 1}, Length[data[[All, 2]]]];
data2 = data;
data2 = {data2[[All, 1]], LowpassFilter[data2[[All, 2]], 0.2]}\[Transpose];
ListLinePlot[{data, data2}, AspectRatio -> 1/4, ImageSize -> 800]

I'm not sure why you want to use a Kalman filter precisely, I think your data is simply high-frequency noise that can be filtered out with a LowpassFilter... Kalman filter needs a theoretical 'model' behind the noise, which can be tough to justify...

Perhaps you can measure multiple periods of a collapsing/oscillating bubble and average it over many periods, that would be ideal. I believe these processes are quite stable over many periods.