Lauren,
These are very clever suggestions and really leverage available ArcGIS tools. However, I would respectably disagree with the statistical tractability of your recommendation.
1) By aggregating the binary data to a counts, represented by each fishnet cell, you have effectively invalidated the null spatial distribution that would indicate clustering. Additionally, since these units are arbitrary, you are adding an additional error component associated with the Modifiable Areal Unit Problem (MAUP). In effect, you are no longer representing the underlying Bernoulli spatial process but rather an arbitrary inhomogeneous intensity process. The assumption of a homogeneous random field does not hold for the LISA statistic in the same way that it does for PPA statistics, but you are recommending changing the underlying spatial process in a way that could profoundly change inference.
2) The way the data is being partitioned you are breaking the Bernoulli distribution and making it impossible to draw inference around the occurrence of the process.
If I had my druthers, I would use a Poisson point process model. In this way you could test competing hypotheses of ancillary generating processes (covariates) rather than just treating it as a pure spatial process with no deterministic characteristics.
MAUP References
Cressie, N.A. (1996) Change of support and the modifiable areal unit problem. Geographical Systems 3(2�??3):159�??180.
Holt, D., D. Steel, M. Tranmer, N. Wrigley (1996) Aggregation and ecological effects in geographically based data. Geographical Analysis 28 (3):244�??261
Openshaw, S. (1983) The modifiable areal unit problem. Norwick: Geo Books. ISBN 0860941345
Wrigley, N. (1995) Revisiting the modifiable areal unit problem and the ecological fallacy. In Cliff, A.D. Diffusing geography: essays for Peter Haggett. The Institute of British Geographers special publications series 31. pp. 123�??181