I personally do not see this time being ideal for your first cycle. Based on your post, you have been dieting (mostly) for the last two years, I would expect your total test levels to be down from that alone. But you always want IMO 2-3 total test readings, while at least in caloric maintenance for sometime prior to starting something like TRT. The total test blood draw should be done after 7-9 hrs of sleep, within 2 hours waking, ideally between 8-10 am.
A test cycle is not likely to move the needle on your goals much IMO. To lose more fat at a good pace, you need to be in a caloric deficit. If you are not losing fat now, that means you really are not there.
To build strength, esp the numbers you have as goals, there are likely 10s to 50s of program alterations that can be made to increase you strength levels outside of simply trying to add more muscle tissue. Yes, adding muscle helps increase the contractile force a muscle so strength can go up. But there is also a neurological side to strength, which means you can simply focus on maintaining the muscle you have or building very slowly in deficit while leveraging neurological adaptations to help push strength up for now.
What your current strength stats on the main lifts and your current routine?
I personally view cycles as an ace card that should be used when the time really comes, or at least, consider that you likely can only do so many as for most, the negative health effects can catch up, so at least pick your cycles and moments with some selectivity.
Just my personal opinion though. You can do as you wish, its your health and life.