OK, here is what I have. I'll skip some derivations (I'll provide them later if you are interested) and just describe the conclusions. The final tables apply if you have noiseless data. Any noticeable amount of noise will cost you quite a bit here.

The problem of how to distinguish between *two fixed* densities $p(x)$ and $q(x)$ is classical. Suppose that we want to bound the combined probability of error by some small $\theta>0$. This means that if we are allowed to take $n$ samples, we have to find some set $E\subset\mathbb R^n$ such that $\int_E P+\int_{E^c}Q\le\theta$ where $P(x_1,\dots,x_n)=p(x_1)\dots p(x_n)$ and similarly for $Q$. Here $E$ is the set where we declare $q$ to be actual density. Note that in no way can this sum be better than $\int\min(P,Q)$ and we can achieve that by the standard maximal likelihood decision: we declare the density $Q$ if $P(X_1,\dots,X_n)<Q(X_1,\dots,X_n)$ and $P$ otherwise. We also can get a fairly clear idea of the necessary sampling size. In fact, we can tell it almost up to a factor of $2$. Note that $\min(P,Q)\le\sqrt{PQ}$, so
$$
\int\min(P,Q)\le \left(\int \sqrt{pq}\right)^n
$$.
On the other hand,
$$
\left(\int \sqrt{pq}\right)^{2n}=\left(\int \sqrt{PQ}\right)^{2}\le
\left(\int\min(P,Q)\right)\left(\int\max(P,Q)\right)\le 2\int\min(P,Q)
$$
Thus, if $\int\sqrt{pq}=e^{-H}$, then to reach the level $\theta$ of combined error, we need at least $\frac 12 H^{-1}\log\frac 1{2\theta}$ and $H^{-1}\log\frac 1\theta$ samples will suffice.

The problem with your case is that we test not two densities but two *families* of densities against each other. However, if my computations are correct, we are lucky and the likelihood test that distinguishes the worst pair is actually universal enough to achieve the level of confidence given by the above $\sqrt{pq}$ estimate. So assuming that $\lambda_q=w$ (so every other $\lambda$ is $\ge 1$), we can define
$p_L(x)=\frac{N-1}N Le^{-Lx}+\frac 1Nwe^{-wx}$, $q(x)=e^{-x}$ where $L=L(N,w)$ is determined from the maximization problem $\int\sqrt{p_Lq}\to\max$ (which in practice is better to pose as $H=\frac 12\int(\sqrt{p_L}-\sqrt q)^2\to\min$), then the corresponding maximal likelihood text works fine and gives a guaranteed bound $\theta$ for each one-sided error whenever the $\sqrt{pq}$ estimate yields the combined error of $\theta$.

I ran a small program to see what sampling sizes it gives for reasonable $w$ and $N$. The table for the sacramental $\theta=0.05$ is below. The lines are $N,L,n$. \phantom{+} is the artifact of the automatic LaTeX style formatting that I was too lazy to disable.
As you can see, with your $10^5$ samples you are just on the edge of "theoretically feasible" for $w=0.5,N=100$ but if you can drop either number, everything gets fairly nice (if *no noise is present*, of course).

I suggest you run a few simulations and see whether it works for you (the "general theory" should be OK, but I could make some stupid mistakes somewhere). Normally, you are getting something like
$$
n=8N^{\frac 1{1-w}}\log\frac {1}{\theta}
$$
as a rule of thumb for choosing the sample size. This is all "the best performance in the worst case" approach. If you actually have more information than you put in the post, that may help push the numbers down a bit :).

Feel free to ask questions but do not expect a quick answer: life is crazy at this end...

w=0.5
100 \phantom{+} 1.009397 186378
90 \phantom{+} 1.010406 155814
80 \phantom{+} 1.011662 127611
70 \phantom{+} 1.013269 101830
60 \phantom{+} 1.015398 78546
50 \phantom{+} 1.018358 57847
40 \phantom{+} 1.022762 39849
30 \phantom{+} 1.030037 24705
20 \phantom{+} 1.044454 12637
w=0.45
100 \phantom{+} 1.010954 89813
90 \phantom{+} 1.012108 75790
80 \phantom{+} 1.013540 62719
70 \phantom{+} 1.015367 50637
60 \phantom{+} 1.017779 39584
50 \phantom{+} 1.021120 29613
40 \phantom{+} 1.026065 20790
30 \phantom{+} 1.034179 13204
20 \phantom{+} 1.050103 6985
w=0.4
100 \phantom{+} 1.012550 45454
90 \phantom{+} 1.013842 38711
80 \phantom{+} 1.015442 32363
70 \phantom{+} 1.017476 26429
60 \phantom{+} 1.020152 20932
50 \phantom{+} 1.023843 15900
40 \phantom{+} 1.029277 11371
30 \phantom{+} 1.038131 7392
20 \phantom{+} 1.055337 4039
w=0.35
100 \phantom{+} 1.014103 24058
90 \phantom{+} 1.015519 20670
80 \phantom{+} 1.017266 17449
70 \phantom{+} 1.019481 14406
60 \phantom{+} 1.022385 11553
50 \phantom{+} 1.026372 8904
40 \phantom{+} 1.032211 6480
30 \phantom{+} 1.041659 4307
20 \phantom{+} 1.059842 2427
w=0.3
100 \phantom{+} 1.015495 13254
90 \phantom{+} 1.017009 11481
80 \phantom{+} 1.018872 9781
70 \phantom{+} 1.021226 8158
60 \phantom{+} 1.024301 6619
50 \phantom{+} 1.028504 5172
40 \phantom{+} 1.034628 3826
30 \phantom{+} 1.044470 2597
20 \phantom{+} 1.063235 1506
w=0.25
100 \phantom{+} 1.016562 7561
90 \phantom{+} 1.018136 6600
80 \phantom{+} 1.020067 5670
70 \phantom{+} 1.022499 4775
60 \phantom{+} 1.025664 3917
50 \phantom{+} 1.029973 3099
40 \phantom{+} 1.036219 2328
30 \phantom{+} 1.046192 1611
20 \phantom{+} 1.065043 960
w=0.2
100 \phantom{+} 1.017079 4443
90 \phantom{+} 1.018656 3906
80 \phantom{+} 1.020587 3382
70 \phantom{+} 1.023011 2873
60 \phantom{+} 1.026156 2380
50 \phantom{+} 1.030419 1906
40 \phantom{+} 1.036571 1453
30 \phantom{+} 1.046335 1024
20 \phantom{+} 1.064644 625
w=0.15
100 \phantom{+} 1.016716 2675
90 \phantom{+} 1.018218 2367
80 \phantom{+} 1.020052 2064
70 \phantom{+} 1.022349 1768
60 \phantom{+} 1.025319 1478
50 \phantom{+} 1.029333 1197
40 \phantom{+} 1.035099 924
30 \phantom{+} 1.044204 662
20 \phantom{+} 1.061157 414
w=0.1
100 \phantom{+} 1.014952 1647
90 \phantom{+} 1.016263 1466
80 \phantom{+} 1.017861 1287
70 \phantom{+} 1.019857 1111
60 \phantom{+} 1.022433 937
50 \phantom{+} 1.025903 766
40 \phantom{+} 1.030871 599
30 \phantom{+} 1.038684 436
20 \phantom{+} 1.053150 278
w=0.05
100 \phantom{+} 1.010786 1099
90 \phantom{+} 1.011716 983
80 \phantom{+} 1.012849 868
70 \phantom{+} 1.014262 754
60 \phantom{+} 1.016083 641
50 \phantom{+} 1.018533 528
40 \phantom{+} 1.022034 417
30 \phantom{+} 1.027529 308
20 \phantom{+} 1.037677 200

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