Configuring IMU FIS1100
Thanks for adding the schematic and the answers so far. Based on that, I have some concerns about the hardware design, and some suggestions for you to make progress with your troubleshooting.

Hardware:

a) By you reading register 0 of the FIS1100 successfully, that gives me some confidence in the I2C bus functionality. However:


Even though the I2C bus is working at least to some extent, personally I would use a low capacitance (i.e. not a "x1") scope probe and check the I2C waveforms on both sides of the PCA9306 I2C voltage translator. This is to get even more confidence in the waveform shape and check that it meets the timing requirements shown in the FIS1100 datasheet.
The need to check the I2C bus waveform is especially true on the 3.3V (MCU) side of the PCA9306, where I am surprised at the designer's choice of 10k\$\small\Omega\$ pull-up resistors, especially if the bus will be operating at more than 100kHz.


b) Although you say that your design is based on the datasheet, it is also different in several ways, including:


No MCU control of the FIS1100 RST (reset) pin; just a power-on reset. Even if that reset pulse meets the FIS1100 requirements (you need to check that), you have no way to reset the FIS1100 without a power-cycle. Also, as I will explain later, the "official" FIS1100 SDK goes through a more complex (but not fully explained) reset procedure which you cannot do, without MCU control of the FIS1100 reset pin.
You tied RESV3 pin to ground via 100k\$\small\Omega\$ resistor. The datasheet says it should be tied to ground, and a direct connection to ground is shown in the datasheet. We don't know how your change to the example design could affect the FIS1100 behaviour, as this is a reserved pin. However unless you have specific information from the IC manufacturer to deviate from the datasheet, such deviation is generally a risky plan.
Based on my experience with other IMUs, good power decoupling is vital for correct behaviour. I see your schematic shows the recommended capacitors C20 and C21, but the PCB layout also needs to be considered.
The recommended decoupling is missing from the PCA9306 VREF2 pin (see section 9.2.2.1 in the PCA9306 datasheet).


Software:

I did a quick search for you, but I only found one place with sample code for using the FIS1100 - and that is the official SDK from the manufacturer. Since you haven't mentioned using that, I will explain how you can get it:


Go to this link to create a "My Fairchild" account. (Obviously that link may rot over time, so be prepared to check their website if that link fails.)
Login to your new account, update your account profile and fill-in the address of your company (my old "My Fairchild" account was missing one of the address fields, and the download would not continue until this part of the account profile was completed).
Go to: https://www.fairchildsemi.com/evaluate/evaluation-boards/FEBFIS1100.html
Download the user guide for their FIS1100 evaluation board from the link on that page; it shows you a sample schematic and PCB layout.
At the bottom of that page under "Related Resources", click the "FEB Supporting Files (ZIP)" icon (I can't link it here, as it uses a bunch of JavaScript and must be clicked from the page I listed above, and you must be logged-in to your "My Fairchild" account first).
After you read and accept the license agreement, you can download the file (current version is 41.2MB) which is a Windows installation executable (unfortunately). When executed (requires admin permissions) it will offer to install all the software (and drivers) needed to use their FIS1100 evaluation kit, on their recommended MCU board.
Under the installation directory e.g. default: C:\Program Files\Fairchild Semiconductor\FIS1100 SDK 1.0.3 look under the SDK directory, including the Documentation, Examples and fis_driver sub-directories. The fis_driver sub-directory source code can easily be reviewed to see how the various FIS1100 registers and signals are used. Due to the software license, I don't believe that I can include that code here.
Interestingly, that source code includes accelerometer and gyroscope self-test routines which don't exactly follow the sequence of events shown in the FIS1100 datasheet. I don't know if the differences are important, but you said that:



  We have also carried out accelerometer self test but there is no
  output data in the output data registers.


Therefore I would consider attempting the self-test again, but following the sequence of register accesses shown in the source code, instead of that shown in the datasheet. It would also be interesting whether the INT2 signal is behaving as described in the datasheet, when you attempt to perform the self-test.


Note that Fis1100_reset() function in the example source code, manually toggles the FIS1100 RST pin (which your schematic doesn't allow you to do), asserting it for 100uS, and then sends some commands for which I can't (quickly) find the meaning.  Again, that might (or might not) be relevant to the incorrect behaviour you are seeing.
You said that you:



  Read STATUS0 register [...] Read Output data registers [...] ctrl7 = 0x83


OK, so you have CTRL7 bit 7 (SyncSimpl) set. Therefore it may be helpful for you to check exactly what is reported in STATUS0 register. For example, if the accelerometer and gyroscope bits are all zero, that would suggest the accelerometer and gyroscope are not producing any readings. It would also be interesting whether the INT2 signal behaves as described in the datasheet when you are trying to do this.

I hope those comments and suggestions are helpful in your troubleshooting - of course it is up to you whether you decide to use any of that information :-)