It would be nice if the photos were somewhere that didn't require
a Google account to access...
The Diamond BB7 and Comet 250 are unlikely to feature on the list of
"absolute best antennas I can deploy" in most circumstances. You may
want to read
G8JNJ's analysis of these antennas.
Note that performance varies depending on whether the antenna is
mounted to a metal mast, and if the mast is grounded or not.
What is your vertical height limit? A quarter wave vertical for 20m,
with elevated radials (even if they have to wrap around the house or
run along the fence) should work pretty well. One for 40m would be
even better, although it is around 32 feet tall: by using a dual-band
matching network at the feedpoint you can use it on both bands.
Rather than a self-supporting vertical, you can try a wire running up
one of the palm trees.
If you have a wood fence, you can use that to run elevated radials
(at least in two directions). Mounting the antenna on the fence then
keeps it further from the house (although closer to the neighbors).
You might need to think of a way to disguise the antenna, like mounting
a bird house on it.
One approach is to use a "screwdriver" remotely tunable mobile
antenna, where the top section can be lowered when not in use so it
isn't as visible. This could then be tuned from, say, 40m through
10m from inside the house, although it would need radials for best
performance.
Hams have used the Hustler and HyGain verticals with just a ground
post for generations. Especially if you have good soil conductivity,
these can work well enough. I suspect your house is stucco, in which
case having the antenna too close to it may make more of a difference
in performance than using less-than-optimal radials.
How tall are your palm trees?
Do you have metal or plastic rain gutters on the house?
What type of roof material do you have?
Is the flag pole easily removable to add an antenna to it?
Or can you add a second one?
There are lots of options, it is just a matter of finding the
one that works best in your specific situation.