Bridgehead carbocation: Violation of Bredt's Rule
The reaction in question is known and has been studied fairly extensively, both experimentally and computationally, and is accepted to go via the bridged carbocation that you propose.  

To give some data, one experiment  measured the relative reaction rate of a series of such structures towards solvolysis in 80% aqueous ethanol (basically the reaction you describe above): 



The intermediate carbocation is known as a non-classical carbocation- a carbocation that does not conform to the standard organic chemistry definition of a planar sp2 intermediate (as we often see with SN1 reactions). 

In the case of the norbornyl cation, the carbocation can be observed spectroscopically by treating the parent halide with a superacid such as $\ce{SbF5.SO2ClF}$ at very low temperatures: 



One issue with non-classical carbocations is their tendency to rearrange, such that you wouldn't get a single product as predicted, but instead end up with a mixture of products resulting from other classical carbocations formed through hydride shifts on the non-classical carbocation (the exact mix of products would depend upon the conditions of the reaction). 

References

All figures taken from Carey, F. Advanced Organic Chemistry - Part A