DEFICIENT ACFT ENGINE PERFORMANCE; HIGH GROSS WEIGHT AND AMBIENT TEMPERATURE CONSPIRE TO LIMIT AA5 FLT CREW'S ABILITY TO OBTAIN REQUIRED IFR ALTITUDES.

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250 NM IFR XCOUNTRY TRAINING FLT (REQUIRED BY FAR 61.65) FOR INST RATING. FLEW ZZZ1 TO ZZZ VIA AIRWAYS/VECTORS. EXECUTED VOR AND LOC BACK COURSE APCHS. PLANNED TO REFUEL/RE-FILE IFR BACK TO ZZZ1. ONLY NEEDED 5 GALLONS OF FUEL. FBO POLICY WAS EITHER $15 LNDG FEE; PURCHASE 15 GALLONS FUEL; OR TOP TANKS. AA-1A ONLY HOLDS 22 GALLONS. ABOUT 90 DEGS F OUTSIDE AIR TEMP. ENG RAN HOT ON CLB FROM ZZZ1; SO WE ONLY WANTED 5 GALLONS. FELT PRESSURED TO TOP OFF BY FBO; SO TOOK ON 9.6 GALLONS. NOW AT/SLIGHTLY OVER MAX GROSS WT AND NEAR EDGE OF PERFORMANCE ENVELOPE. FILED FOR 4000 FT AND PLANNED SHALLOW CLB. AFTER TKOF; ONLY MADE 2000 FT BEFORE ENG OIL TEMP STARTED TOWARD RED LINE. ASKED DEP TO LET US LEVEL AT 2000 FT AND THEN CLB TO 4000 FT LATER. DEP SAID UNABLE TO FLY ZZZ TO ZZZ2 AT 2000 FT DUE TO 2000 FT RADIO TWRS AHEAD. ASKED FOR VECTOR AROUND TWRS. DEP GAVE US HDG BUT ADVISED WE COULD NOT CONTINUE IFR UNLESS WE CLBED TO 3000 FT; OTHERWISE WE HAD TO CANCEL AND PROCEED VFR. CONSULTED WITH OTHER PLT (ACFT OWNER) AND BOTH AGREED WE COULD NOT CLB TILL WE BURNED OFF SOME FUEL. RELUCTANTLY CANCELED IFR; WHICH MEANT WE HAD TO RE-ACCOMPLISH THE WHOLE FLT AGAIN; SINCE FAR 61.55 REQUIRES 250 NM XCOUNTRY UNDER IFR. ASKED FOR VECTOR TO ZZZ1. DEP CTL GAVE US VECTOR AROUND RESTR AREA. REVIEWED THE IFR CHART AGAIN; AND MEA WAS; AS I THOUGHT; 2200 FT MSL (TOO LOW FOR A 2000 FT TWR). I CALLED DEP AND ASKED IF I FILED A NEW IFR FLT PLAN BACK IN SYS AND CLRED US AS REQUESTED WITH DIRECTION TO CLB TO 3000 FT WHEN ABLE. WE WERE ABLE TO REACH 3000 FT ABOUT 30 OR SO MI NE OF ZZZ2. AFTER HDOF TO APCH; ASKED FOR 3000 FT AS FINAL. WE FLEW 3000 FT TILL ABOUT 30 MI S OF ZZZ1 WHEN ZZZ1 APCH ASKED FOR CLB TO 4000 FT FOR TERRAIN -- A MYSTERY; AS MEA WAS 2500 FT. WE SAID WE'D TRY TO CLB BUT NOT MIGHT MAKE 4000 FT. CTLR SAID STAY AT 3000 FT; WHICH WE HELD ALL THE WAY TO GS INTERCEPT FOR ILS. 2 LESSONS: 1) DON'T LET THE FBO CTL YOUR FUEL LOAD; AND 2) KEEP WORKING WITH ATC FOR A SOLUTION. FORTUNATELY; THE CTLR DECIDED TO HELP.